Yang-Mills Configurations on Nearly K¨Ahler and G2

Gottfried Wilhelm Leibniz Universitat¨ Hannover Mathematisch-physikalische Fakultat¨ Institut fur¨ Theoretische Physik

Diploma Thesis

Yang-Mills conﬁgurations on nearly Kahler¨ and G2-manifolds

Irina Bauer

Date of birth: November 28th 1985, Tomsk

Supervisors: Prof. Dr. Olaf Lechtenfeld and Prof. Dr. Alexander D. Popov

1st referee: Prof. Dr. Olaf Lechtenfeld 2nd referee: Prof. Dr. Norbert Dragon

Hannover, Mai 19th 2010

Erklarung¨

Hiermit versichere ich, dass ich die Diplomarbeit selbstandig¨ und lediglich unter Benutzung der angegebenen Quellen und Hilfsmittel verfasst habe.

Hannover, 19.05.2010

Acknowledgements

In the first place I thank my advisor Prof. Dr. Olaf Lechtenfeld for giving me the opportunity to be a diploma student in his group, for continuous support from his side during the whole period of the past twelve months and friendly atmosphere. Of course I also thank the Institut fur¨ Theoretische Physik for offering the diploma program. A special thanks goes to my co-advisor, Prof. Dr. Alexander D. Popov, who is most responsible for helping me to organize the entire process of writing of this thesis and to complete it in an adequately way as well as for giving me an outlook of needed mathematical basics and for answering different questions concerning them. He taught me how to write academic papers, spend a lot of time correcting my drafts of the thesis and supported me by discussing with me different topics and problems arising while writing the thesis. I am also thankful to Felix Lubbe for discussing and comparing several calculations and results. Moreover, I also want to thank Dr. Tatiana A. Ivanova for carefully analyzing and checking my drafts for mistakes and giving useful remarks. To the whole group consisting of persons just mentioned I want to express gratitude for a mutual support and a good teamwork. Finally I thank my family, which always supported me during my study of physics and encouraged me in pursuing my interests.

Contents

Contents iii

1 Introduction1

2 Mathematical preliminaries7 2.1 Metric structure on vector spaces...... 7 2.1.1 Vector space Rn ...... 7 2.1.2 Coordinates on Rn ...... 8 2.1.3 Linear transformation of Rn ...... 8 2.1.4 Euclidean metric on Rn ...... 9 2.1.5 Metric of signature (p, q) on Rn ...... 9 2.1.6 Groups O(p, q) and SO(p, q) ...... 10 2.1.7 Translations and group IO(p, q)...... 10 2.1.8 Conformal group on Rp,q ...... 11 2.1.9 Rp,q as a topological space...... 11 2.2 Symplectic structures on R2n ...... 12 2.2.1 Symplectic form ω on the space R2n ...... 12 2.2.2 Transformation group Sp(2n, R) preserving ω ...... 12 2.2.3 Degenerate and non-degenerate symplectic forms on R2n . 13 2.3 Complex structure on R2n ...... 13 2.3.1 Complex coordinates on R2n ...... 13 2.3.2 A complex structure J on R2n ...... 14 2.3.3 Subgroup GL(n, C) of GL(2n, R) as a group preserving the complex structure J ...... 14 2.3.4 Complex structure J on the space R2p,2q and Hermitian metric g of signature (p, q)...... 15 2.3.5 The group U(p, q) preserving J and g ...... 16 2.3.6 Euclidean case as a subcase...... 17 2.3.7 Symplectic structure ω on R2n and R2p,2q ...... 17 2.3.8 U(p, q) as a subgroup of Sp(2n, R)...... 18

iii iv CONTENTS

2.4 Algebraic structures on vector spaces...... 18 2.4.1 General notion of algebra...... 18 2.4.2 Associative and non-associative algebras...... 20 2.4.3 Algebra of quaternions H ...... 20 2.4.4 The group Sp(1) SU(2) as a group preserving the quaternion structure...... 20 2.4.5 Vector space Hn and the group Sp(n) ...... 23 2.4.6 Metric of signature (p, q) with p + q = n on Hn and the group Sp(p, q)...... 23 2.4.7 Algebra O of octonions ...... 24 2.4.8 Properties of the octonion structure constants ...... 26

2.4.9 Group G2 and Spin(7) related with the octonion algebra . 26 2.4.10 Grassmann algebra: definition and main properties . . . . 27 2.4.11 Clifford algebra of the vector space Rp,q of signature (p, q) . 29 2.4.12 Generators of Clifford algebra...... 31 2.4.13 The 8-fold periodicity of Clifford algebra...... 32 2.4.14 Matrix representations of Clifford algebra over C and R . 34 2.4.15 Spinors (fermions) as representation spaces of Clifford algebras ...... 36 2.4.16 Group Spin(p, q)...... 38 2.4.17 Supersymmetry and superalgebra...... 39 2.5 Theory of general Lie algebras...... 40 2.5.1 Lie algebras...... 41 2.5.2 Matrix Lie algebras...... 41 2.5.3 Structure constants, Bianchi identities, Killing-Cartan metric 43 2.5.4 Roots and weights ...... 44 2.5.5 Some elements of representation theory...... 48 2.5.6 Definitions of affine and Kac-Moody algebras ...... 48 2.6 Elements of Lie group theory ...... 51 2.6.1 Matrix Lie groups and their relation with Lie algebras . . . 51 2.6.2 Generic definition of groups...... 53 2.6.3 Topological groups and smooth Lie groups...... 53 2.6.4 Left, right and adjoint action of a Lie group G on itself . . 54 2.7 Homogeneous spaces...... 55 2.7.1 Closed subgroup H of a Lie group G ...... 55 2.7.2 Homogeneous spaces G/H and H G of a Lie group as \ equivalence classes...... 55 2.7.3 Splitting of the Lie algebra Lie G = Lie H (Lie H) . . . . 56 ⊕ ⊥ 2.7.4 Splitting of commutators ...... 57 2.7.5 The Maurer-Cartan equations...... 58 2.7.6 Metric, Levi-Civita connection, affine connection and torsion 58 CONTENTS v

2.7.7 Kahler¨ coset spaces G/H ...... 61 2.7.8 Nearly Kahler¨ coset spaces in d = 6...... 62

3 Yang-Mills-theory on nearly Kählermanifolds 65 3.1 Yang-Mills equation on manifold M ...... 65 3.1.1 Principal fibre bundle ...... 65 3.1.2 Transition functions ...... 66 3.1.3 Associated fibre bundle ...... 66 3.1.4 Associated vector bundle ...... 67 3.1.5 Connection ...... 67 3.1.6 Holonomy group...... 68 3.1.7 Connection one-form...... 68 3.1.8 Curvature 2-form...... 69 3.1.9 Bianchi equation ...... 70 3.2 Yang-Mills equations on G/H ...... 71 3.2.1 Coset space G/H ...... 71 3.2.2 Connection on G/H ...... 72 3.2.3 Yang-Mills equations...... 72 3.2.4 Hermitian-Yang-Mills equations...... 73 3.2.5 Some explicit solutions of the Hermitian-Yang-Mills equations ...... 74

3.3 Manifolds with G2-structure...... 77 3.3.1 Seven-dimensional manifolds with G2-structure ...... 77 3.3.2 Cones over nearly Kahler¨ 6-manifolds...... 78

3.3.3 Special G2-manifolds of type R G/H with nearly Kahler¨ × coset spaces G/H ...... 79

3.4 Yang-Mills on G2-manifolds R G/H ...... 81 × 3.4.1 Yang-Millsequations on spaces R G/H: G-invariant ansatze¨ × and reductions to kink-type equations...... 81 3.4.2 General solution of the G-invariance condition ...... 83

3.4.3 Calculation of traces over IAIB ...... 86 3.4.4 Yang-Mills on R SU(3)/U(1) U(1) ...... 88 × × 3.4.5 Yang-Mills on R Sp(2)/Sp(1) U(1) ...... 90 × × 3.4.6 Action ...... 94 3.4.7 Action for the case R SU(3)/U(1) U(1)...... 97 × × 3.4.8 Action for the case R Sp(2)/Sp(1) U(1)...... 98 × × 3.4.9 Yang-Mills on R G2/SU(3)...... 99 × 3.4.10 Yang-Mills on R S3 S3 ...... 100 × × 3.4.11 Solutions on R SU(3)/U(1) U(1) ...... 100 × × 3.4.12 Solutions on R Sp(2)/Sp(1) U(1) ...... 105 × × 3.4.13 Instanton-anti-instanton chains and dyons ...... 106 vi CONTENTS

A Appendix 109 A.1 Jacobi identities...... 109 A.2 Jacobi elliptic functions...... 109

List of Tables 111

Bibliography 113 Chapter 1

Introduction

It was noticed a long time ago that the physical structures of the visible world we are living in can elegantly be described by using a mathematical framework. Es- pecially the Einstein’s vision of space-time as a dynamical object of Riemannian geometry demonstrates how the mathematical understanding of physical pro- cesses helps in logical admission of complicated lows which governs all observed physical phenomena. In generally, it is very exiting to see how just theoretical initially apparently useless mathematical models can once be applied in physics. If we look at the existing physical theories, we see that while the combination of the “Standard Model”, a specific four-dimensional quantum field theory based on Yang-Mills theory, and general relativity makes it difficult to describe some phenomena (for example physics at the energy over the Plank scale M 1019 pl ∼ GeV, “coupling unification” and the “hierarchy problem”), string, superstring and more general M-theory are mathematical models which at present are the best candidates for a unification of quantum theory of all matter and interactions with gravity. Although there is still missing a theory with a geometrical interpretation, which one actually would expect from a new concept, they produces various remarkable results at the frontier between mathematics and physics. Historically, it was one of the achievements of the last century attained in accelerator experiments which allowed to unify the three fundamental forces of nature, namely electromagnetism, the weak interaction, and the strong interaction in one gauge theory SU(3) SU(2) U(1), showing that quaks and W- × × bosons as well as photons form representations of the gauge groups SU(3) and SU(2) U(1), respectively. The Standard Model based on this knowledge has × therefore a special meaning in the constructions of new theories. One takes it for granted that additional structures of string theory, M theory or whatever candidate fundamental theory must lie beyond the Standard Model. In string theory this prerequisite is described by configurations of type M Xd based 4 ×

1 2 CHAPTER 1. INTRODUCTION

on a choice of inner structure of the space-time M4 given by a compact manifold Xd. The latter furnishes the space-time with d extra dimensions as well as remarkable features. In physics this mechanism became the name compactification or Kaluza-Klein theory. It is clear that independently of what kind of interior structure would be preferred, this additional setup is preserved by the gauge symmetry, corresponding to the isometry group of global transformations of the manifold, which must be contained in the SU(3) SU(2) U(1)-symmetry × × in some sense in order to lead to the Standard Model. The best auxiliary by the description and handling of this topic is of course the language of differential geometry. In any case one of the most important aspects of the theory is the type of the internal compact manifold, which has to be of a very small size and is required to yield dimensional reduction of the full theory to the well-known theory on four dimensional space-time. As to string theory, born in the 70’s and became a large extent after 1984, today the research in string theoretical field focuses on the recently most interesting areas such as compactification on special holonomy manifolds and approaches to mirror symmetry. However, it is still a central problem to derive the particular effective field theory, the Standard Model, from string theory. But nevertheless string theory successfully and above all naturally solves some of the problems of the Standard Model such as the ultraviolet divergences and the inclusion of gravity in the theory, which is impossible in the pure quantum field theory of the Standard Model as well in its generalizations GUT’s due to the renormalization problem. In particular, combination of string theory with supersymmetry and its way of thinking of the elementary particles as of the extended objects (strings) smoothes singularities of the theory and appends additional vibration excitations interpreted as spin-2 particles - gravitons. In general, supersymmetry is a necessary ingredient of a fundamental theory. Since all particles in nature are distributed in two groups - fermions and bosons, the theory must contain both of these types. When we include fermions in string theory, supersymmetry which relates each bosonic particle a fermionic super- partner arises automatically so that fermions and bosons are grouped pairwise together into supermultiplets via supersymmetry. One of the conditions to the theory of superstrings to be consistent and especially free from anomalies is to fix the dimension number D = 4 + d = 10. Out of it results the above-mentioned necessary compactification, i.e., curling up of the extra 6 dimensions into a small compact submanifold of the physical space which can not be detected in the real 33 life (the string scale is about 10− cm). In terms of weak coupling perturbation theory there exist only five different consistent superstring theories known as Type I SO(32), SO(32) Heterotic, E8 E8 Heterotic, Type IIA and Type IIB. The × first three of them have only one ( = 1) supersymmetry in ten dimensions, N while the last two have = 2. N 3

As it turns out this fact narrows the variety of possible ways of comactifi- cations of the 10-dimensional superstring theory on a 6-dimensional compact manifold in a very strong manner, because the most trivial compactifications such as curling up of the 6 extra dimension to circles would preserve too much supersymmetries. In order to impose only one supersymmetry in 4 dimensions, which one predicts at the energy scales above 1 TeV, we can compactify on a special kind of 6-manifold, namely on Calabi-Yau manifolds with SU(3)-holonomy. However, the multiplicity of different Calabi-Yau manifolds suggests the possibility of arising of too many various 4-dimensional theories. Furthermore, Calabi-Yau compactifications cause many massless moduli fields in the resulting four-dimensional effective theories (in general, massless moduli fields, as intrinsic components of effective theories below the Planck scale, exist in all known superstring theories which parameterize continuous vacuum-state degeneracies [1]). One possibility to solve this so-called moduli-stabilization problem is to allow for non-trivial p-form fluxes on Xd. String vacua with p-form fields along the extra dimensions, called flux compactifications, relax the restriction of the internal space to be Calabi-Yau demanding only presence of a SU(3)-structure on the internal manifold.Besides that, supersymmetry-preserving compactifications on d 10 d space-times of the form M10 d X with further reduction to M − requires − × validity of the first-order BPS-type (Bogomol’nyi-Prasad-Sommerfeld) gauge equations on Xd (see [2] and also references therein), in particular, generalization of the Yang-Mills anti-self-duality equations in four to higher-dimensional manifolds with a special holonomy. Furthermore, it is natural in six dimensions, d = 6, that there appears a 2-form on X6 which we can use to define a three-form considered as torsion. So we H may search for an appropriate compact manifold Xd having a non-vanishing torsion field. Some of the rear candidates for the internal space X6 with torsionful inner geometry in the context of heterotic string theories are the nearly Kahler¨ manifolds which especially carry a natural almost not integrable complex structure and are contained in a class of coset spaces referred to as non-symmetric coset spaces. In this diploma thesis we are going to derive non-perturbative instanton solutions to the equations of motion of Lie G-valued Yang-Mills fields on the space R G/H, where G/H is a compact nearly Kahler¨ six-dimensional homoge- × neous space and the manifold R G/H carries a G2-structure. Initially instantons × were found as topologically non-trivial solutions of the duality equations of the Euclidean Yang-Mills theory with finite action [3], i.e., as soliton solutions of Yang-Mills equations localized in space and time. The first such solutions were discovered in the case of four-dimensional Euclidean space compactified to the four-dimensional sphere and earned the names pseudoparticle and instanton. In four Euclidean dimensions instantons are non-perturbative BPS configurations 4 CHAPTER 1. INTRODUCTION solving the first-oder anti-self-duality equation and forming a subset of solutions to the full Yang-Mills equations. In higher dimensions they can also be found as solutions of BPS configurations known as generalized anti-self-duality equations or Σ-anti-self-duality already mentioned above. One of the aspects associated to the instantons is that they can be understood as tunneling particles characterized by trajectories connecting degenerated minima of the potential energy or in other words of the Yang-Mills vacuum. Therefore, in this thesis we are going to investigate the Yang-Mills theory on the only four known examples of compact nearly Kahler¨ 6-manifolds 6 3 3 3 SU(3)/U(1) U(1), Sp(2)/Sp(1) U(1), G2/SU(3)= S and SU(2) /SU(2) = S S , × × × which are in particular non-symmetric reductive and homogeneous. Further we consider the corresponding equations of motion on spaces of the form R G/H, × where G/H stands for one of the coset spaces just listed, and endow it with a G2-structure. One of the future perspectives could be in the extension of the found solutions on this simplified model to solutions on the Minkowski spaces of the form R4 G/H. × Firstly, we introduce the general ansatz of a Yang-Mills theory with gauge group G over the base space R G/H. For all four cases these ansatze¨ are G- × equivariant which performs the dimensional reduction and cancels a part of the additional coset variables. The gauge potential of the theory is given by a connection on an associated principal bundle parameterized by complex functions φi, i = 1, 2, 3. In this case we are able to write down the ansatz in general form without any knowledge about the explicit type of the coset. From the G-equivariant ansatz for the gauge potential we derive the corresponding field strength and the corresponding Yang-Mills equations. Then we distinguish between the four cases of different coset spaces and analyze specifying the Yang-Mills equations at first on R SU(3)/U(1) U(1). Our ansatz reduces the Yang-Mills equations to × × coupled Newtons equations of type φ¨ i = f (φ1, φ2, φ3) describing a particle in complex three-dimensional space under influence of a cubic force f . We also show the symmetry characteristic to the equations. Then we go over to the same consideration of the case R Sp(2)/Sp(1) U(1) seeking for common relations be- × × tween the two cases. Furthermore, we give for each case the explicit form of the action functional, the found equation of motions can be derived from, and de- termine the quartic potential V. Then we construct for specific values of torsion some BPS-type solutions coming either from differential gradient or Hamilto- nian flow equations and yielding instanton configurations of finite action. By a duality transformation, which relates solutions for different values of torsion, infinite-action solutions to the Yang-Mills equations are presented as well. At last we give some non-BPS solutions on G2/SU(3) such as instanton-anti-instanton solutions (sphalerons) and dyonic solutions (bounces). They can be lifted up to the solutions on SU(3)/U(1) U(1) and Sp(2)/Sp(1) U(1). × × 5

The structure of thesis has the following order. The content is divided into two main chapters. Chapter 2 is a purely mathematical part of the thesis and consists of a detailed introduction to the basic notions of algebra, especially, it efforts an outlook about the major structure groups and corresponding invariant tensor fields, algebraic structure such as Clifford algebras, Lie algebras and related Lie groups. It also has a look at the theory of homogeneous spaces, specifically of nearly Kahler¨ coset spaces having a SU(3) structure. To understand the latter we are going to introduce differential structures on manifolds, e.g., almost complex structure, metric, connection, torsion etc. Chapter 3 is dedicated to the dealing with manifolds specified to be nearly Kahler.¨ Before consideration of Yang-Mills theories on this spaces we need to understand additional structures given by fibre bundle, principal bundle and the generalized notions from Chapter 2, i.e., connections and curvature on such bundles. Further, we introduce the definition of the Yang-Mills theory on coset spaces and give the notion of a G2-structure on 7-dimensional manifolds specializing in the case of cylinder spaces over 6- dimensional nearly Kahler¨ coset spaces. The rest of Chapter 3 is dedicated to the explicit derivation of the Yang-Mills equations on nearly Kahler¨ spaces as well as to the finding of their solutions according to the conditions described above.

Chapter 2

Mathematical preliminaries

During the complete chapter we want to provide the concept of basic mathematical structures needed as a background knowledge for understanding of the Yang-Mills theory. We start with an introduction to fundamental notions on vector spaces and subsequently go over to algebraic, group theoretical and geometrical deﬁnitions. After that we will use the theoretical approach established in the ﬁrst part of the chapter to deepen in the topic specifying and concretizing the subject in the theory of nearly Kahler¨ coset spaces.

2.1 Metric structure on vector spaces

In this section we introduce the vector space as a fundamental object and make it clear that any structure on it corresponds to some group of transformations. In this way every structure can be interpreted as an invariant of a certain group preserving it.

2.1.1 Vector space Rn

Let R denote the ﬁeld of real numbers. The system of all possible ordered n-tuples x = (x1, x2, ..., xn), where xi R for each i = 1, ..., n, deﬁnes an n-dimensional ∈ Euclidean space Rn. Each point x Rn is determined by its corresponding ∈ n-tuple (Cartesian coordinates) in a unique way, i.e., two points x, y Rn are ∈ equal if and only if their coordinates coincide: xi = yi. We introduce on Rn the following vector operations:

x + y = (x1 + y1, ..., xn + yn), (2.1) c x = (c x1, ..., c xn) (2.2) · · · 7 8 CHAPTER 2. MATHEMATICAL PRELIMINARIES for two given points x, y Rn and c R. With the sum and the multiplication on ∈ ∈ scalars the Euclidean space acquires the structure of n-dimensional vector space. The simplest example is the line of real numbers which can be described by only one coordinate x1, i.e., its dimension is 1. The plane R2 can be furnished with two coordinates x1, x2 and the ordinary space R3 with three; they are 2- and 3-dimensional spaces, respectively.

2.1.2 Coordinates on Rn

In order to describe an arbitrary point x of Rn one can always write it in terms of its coordinates. The most natural way to do this is to represent it as linear combination of the standard basis vectors ei , i = 1, ..., n: { } Xn 1 n i x = (x , ..., x ) = x ei, (2.3) i=1 where e1 = (1, 0, ..., 0), e2 = (0, 1, 0, ..., 0), ..., en = (0, ..., 0, 1). In particular, this construction gives rise to the identiﬁcation of the vector with its coordinates, i.e., with the coeﬃcients of ei in the decomposition relating to standard basis. This choice of basis vectors is quite standard. In general case, one uses an arbitrary n-vector tuple as a basis up to a point that they are linear independent or, in other words, the basis allows us to construct every vector of Rn.

2.1.3 Linear transformation of Rn

As it was mentioned above, there are many possible ways to associate a basis to n R . Suppose two diﬀerent sets of basis vectors are given: ei and e . We may { } { 0j} write every vector e in terms of the other basis vectors ei : i0 { } n X j ei0 = ai ej, (2.4) i=1 j j i = 1, ...., n and some real valued constants a Rn. The matrix A = (a ) ∀ i ∈ i corresponds to a transformation of the coordinate system, but can be considered as a transformation of the vector space Rn itself. This transformation is obviously linear and maps Rn on Rn bijectively, i.e., it maps an n-basis to another n- basis. On the other hand, considering such transformations as transformations of Rn, one associates smoothly every region of Rn in a one-to-one and onto manner to another region with corresponding coordinates. In this sense all possible transformations form an automorphism group of Rn. One can easily show that the corresponding transformation matrix A is always invertible, since bijective, and thus non-singular, det A , 0. The matrix product leaves the non- singularity unaﬀected. Therefore, equipped with the matrix multiplication, the n n-matrices A build a group. We call it the general linear group GL(n, R) × [4,5,6]. 2.1. METRIC STRUCTURE ON VECTOR SPACES 9

2.1.4 Euclidean metric on Rn

The Euclidean metric is the function Rn Rn R that assigns to any two points × −→ x = (x1, ..., xn) and y = (y1, ..., yn) on Euclidean space the distance between them. This is given by q d(x, y):= x y = (x1 y1)2 + ... + (xn yn)2. (2.5) | − | − − The Euclidean metric on Rn is the standard metric on this space, i.e., it is a positive deﬁnite, symmetric bilinear function, which obeys the triangle inequality. Shortly we can write these constraints as

1. d(x, x) = 0; d(x, y) > 0, (x , y);

2. d(x, y) = d(y, x);

3. d(x, y) 6 d(x, z) + d(z, y), for all z Rn. ∈ It is straightforward to show that the Euclidean metric is symmetric, non- negative and d(x, y) = 0 if and only if x = y. The proof of the triangle inequality makes use of the Cauchy-Schwarz inequality. Relative to the standard basis the Euclidean metric on Rn is given by the matrix   1 0 0    ··· .   .   0 1 .  (gij):= (δij) = diag(1,..., 1) =   (2.6)  . ..   . .     0 1  ······ The Euclidean metric tensor g assigns to every pair of points x, y Rn a scalar ∈ product given by non-degenerate bilinear symmetric map:

g(x, y) = x1 y1 + ... + xn yn. (2.7) p So that we may write d(x, y) = g(x y, x y). The squared inﬁnitesimal dis- − − tance between two inﬁnitely closed points x and x + dx then is

2 2 i j ds := d (x, x + dx) = gijdx dx , (2.8) where Einstein summation convention has been used.

2.1.5 Metric of signature (p, q) on Rn Let us now omit the condition of positive definiteness in the definition of metric. Thus, we say that Rn is endowed with a metric of signature (p, q) if on the space Rn with respect to an arbitrary chosen basis a smooth bilinear non-degenerate symmetric function g : Rn Rn R is defined, whose representation matrix × → 10 CHAPTER 2. MATHEMATICAL PRELIMINARIES

(gij) has p positive and q negative eigenvalues (p + q = n). Since the matrix (gij) is symmetric and non-degenerate it can be diagonalized in a way that the Pp Pq metric can be written as: g(x, y) = xi yi xj yj. In this case we call the i=1 − j=1 metric g either pseudo-Euclidean or simply metric of type (p, q). Riemannian metrics are examples of metrics with a positive deﬁnite signature (n, 0). Whereas a Lorentzian metric is one with signature (1, n 1) or (n 1, 1). − −

2.1.6 Groups O(p, q) and SO(p, q)

Let us now consider linear coordinate transformations of the space Rn, which ﬁx the origin, but also leave every non-degenerate, symmetric bilinear form g of signature (p, q) invariant (p + q = n). These transformations form a group, the pseudo-orthogonal group O(p,q), which in case, if p = 0 or q = 0, contains rotations and reﬂections of the Euclidean space. Since the orthogonal group preserves any metric on Rn, it in particular preserves angles1 between the basis vectors and consequently each matrix A ∈ O(p, q) maps an orthonormal basis to an orthogonal basis. This fact gives 1 T the matrices A O(p, q) the special property: A = Idp qA Idp q, where Idp q ∈ − , , , is the identity matrix of signature (p, q). To see this we consider O(p, q) as a group of transformations preserving the pseudo-Euclidean scalar product i i 1 1 p q n n < x, y >= gijx y = x y + ... + x y ... x y (relative to an appropriate ba- − − sis), i.e., it has to satisfy:

< Ax, Ay > = < x, y >, x, y Rn or (2.9) ∀ ∈ T A gA = g = (gij). (2.10)

Relative to an appropriate basis, g is a unit diagonal matrix Idp,q of signature 1 T (p, q). Hence, A− =Idp,qA Idp,q. In the simplest case, if p = 0 or q = 0, this 1 T implies that A− = A . Also we can conclude that the dimension of O(p, q) is n(n 1)/2. Moreover, one sees that elements of the coset space GL(p+q, R)/O(p, q) − describe all possible metrics on Rp,q by the mapping [A] AT gA, where [A] is 7−→ the equivalence class of A GL(p + q, R) and g the standard unit diagonal metric ∈ on Rp,q of signature (p, q). The special orthogonal group, SO(p, q), is the subgroup of O(p, q) consisting of all elements with the determinant 1.

2.1.7 Translations and group IO(p, q)

Extending the notion of linear coordinate transformation S, we look at transformations of Rn preserving the metric which additionally includes transitions of

1The angle between two vectors x, y Rn is deﬁned via cos α(x, y) = . ∈ √ √ 2.1. METRIC STRUCTURE ON VECTOR SPACES 11 the origin. Then in general we can write the coordinate transformations in the following form

i j i i i i x0 = x S + a , S O(n) a R. (2.11) j j ∈ ∈ Together with the group of orthogonal transformations O(p, q) transitions deﬁne the indeﬁnite orthogonal group IO(p, q). These transformations form the largest group preserving the metric. Its elements are called isometries of Rn. A special meaning in mathematical physics has the group IO(1, n 1) which one calls the − 1,n 1 Poincare group. It is the group of isometries of Minkowski spacetime R − .

2.1.8 Conformal group on Rp,q

We have already note that the orthogonal group preserves angles. Now we turn to the largest group of transformations which has this property. It is called the conformal group of Rp,q. One can show that it can be deﬁned as transformations leaving the metric invariant up to a scalar factor: g(x) Ω2 g(x). It is the 7−→ minimal group that contains the Poincare group as well as the inversion x x . 7−→ x2 It is isomorphic to the group SO(p + 1, q + 1), of which the rotation group SO(p, q) forms a subgroup. The conformal group consists of 4 sorts of transformations. In coordinate representation they have the following forms. Translations: xi x i = xi + ai, ai R. 7−→ 0 ∈ Rotations/boosts: xi x i = xjΛi , where Λ is an element of the Lorentz group. 7−→ 0 j i i i Dilatation: x x0 = λx , where λ R. 7−→ ∈ i i 2 i i x b x Rp,q Special conformal transformations: x x0 = 1 b−x+b2x2 , where b . −→ − · ∈ 2.1.9 Rp,q as a topological space

In order to go over later to the notion of a manifold we introduce the deﬁnition of a topological space.

Deﬁnition 1. : A topological space is a pair (X, I) consisting of the set of points x X ∈ and some family of subsets of X, Ui i I = I, satisfying the following conditions: { } ∈ 1. The empty set and X are in I.

2. The union of any collection of sets in I is also in I.

3. The intersection of any ﬁnite collection of sets in I is also in I.

The elements of I are called the open sets of space X and their complements in X are called the closed sets. The collection of open sets of I is called a topology on X. Some family Bj j J = J is called a basis of the topology (X, I) if every { } ∈ set of I is a union of some sets of J. In general, a given set X may have many diﬀerent topologies. But there is a standard way to associate a topology to 12 CHAPTER 2. MATHEMATICAL PRELIMINARIES the Euclidean spaces Rn. For the Euclidean space equipped with the standard metric d the open sets are deﬁned as a set of open balls with radius R > 0: B(x, R) = y Rn d(x, y) < R . In particular, this means that a set is open if there { ∈ | } exists an open ball of non-zero radius around every point in the set.

2.2 Symplectic structures on R2n

Continuing our discussion about basic transformation groups, we turn to a little more complicated form, namely the symplectic form, which in particular will later play not less meaningful role than the metric.

2.2.1 Symplectic form ω on the space R2n A symplectic form on the vector space Rn is a map ω : Rn Rn Rn, which is × −→ bilinear, skew-symmetric and non-degenerate, i.e, ω satisﬁes:

n ω(λx + µx2, y) = λω(x , y) + µω(x2, y), x , x2, y R and λ, µ R; • 1 1 ∀ 1 ∈ ∈ ω(x, y) = ω(y, x), x, y Rn; • − ∀ ∈ ω(x, y) = 0, x implies that y = 0. • ∀ In a ﬁxed basis, ω can be represented by a matrix. One can show that non- degenerate symplectic form on Rn exists for n < only if n is even, since every ∞ skew-symmetric matrix of odd size has determinant zero. Typically ω is chosen to be the block matrix ! 0 Id ω = n , (2.12) Idn 0 − 2n where Idn is the n n identity matrix. Any symplectic form on R looks like × that after choosing a suitable basis. In this basis the form ω can be written as Xn ω(x, y) = xi yi+n xn+i yi, (2.13) − i=0 where x, y R2n [7,5,8,9,7, 10]. ∈ For further applications we note that locally any symplectic manifold re- sembles this simple one. For the proof of this proposition, which is known as Darboux’s theorem, see e.g. [6].

2.2.2 Transformation group Sp(2n, R) preserving ω Suppose R2n carries a symplectic form ω. A linear map f : Rn R2n R2n is × −→ called a symplectic map if it preserves the symplectic form, that is, if

ω( f (x), f (y)) = ω(x, y), x, y R2n. (2.14) ∀ ∈ 2.3. COMPLEX STRUCTURE ON R2N 13

The set of all such linear transformations f forms the symplectic group Sp(2n, R), which is in particular a (matrix) Lie group. Any symplectic form ω on R2n looks like the standard symplectic form after 2n choosing a suitable basis. Let ei be the standard basis of R . Then we have { } ! 0 Idn (ω(ei, ej)) = =: J (2.15) Idn 0 − T 2 2n and the matrix J satisﬁes J = J, J = Id2n in R . Using J in terms of the − − standard inner product on R2n we have ω(x, y) =< x, Jy >. Thus, A is a matrix of a symplectic transformation iﬀ < x, Jy >=< Ax, JAy >=< x, AT JAy >. Hence, A Sp(2n) if and only if AT JA = J. Therefore, we can ∈ conclude that the coset space GL(2n, R)/Sp(2n) gives all symplectic forms on R2n by the correspondence [A] ATωA, where [A] is the equivalence class of 7−→ A GL(2n, R) and ω is the standard symplectic from on R2n [5,9,7]. ∈

2.2.3 Degenerate and non-degenerate symplectic forms on R2n

In the definition of symplectic form ω it is possible to relax the condition of non-degeneracy assuming that there exists some x R2n, x , 0, such that ω(x, y) ∈ vanishes for every y R2n. In fact, this means that the function from R2n to R2n ∈ given by f : x ω(x, y) is not an isomorphism anymore or, in another words, 7−→ the kernel of the map is non-trivial. Then the determinant of the associated matrix of ω is zero. In this situation the symplectic form is called degenerate. Returning to the previous definition, we note that the above map f for a non-degenerate symplectic form defines an isomorphism. The determinant of the associated matrix is then non-zero and the associated matrix is not singular. In particular, in any way both statements for degenerate and for non-degenerate symplectic forms are independent of the chosen basis [5].

2.3 Complex structure on R2n

Until now we considered the simplest case of vector spaces, i.e., one built on a ﬁeld of real numbers, but now consider more general notion of complex vector spaces. We will see that they are highly related.

2.3.1 Complex coordinates on R2n

Many geometrical problems are most conveniently formulated and solved by using complex numbers. Also the real vector space R2n can be identiﬁed with a complex vector space. The most simple way to make it clear is to postulate on R2n complex coordinates. We do that as follows. At ﬁrst we might write any 14 CHAPTER 2. MATHEMATICAL PRELIMINARIES point on z R2n in the unique form ∈ i i+n z = x ei + y ei+n, (2.16)

i i+n 2n where i = 1, ..., n, x , y R and em , m = 1, ..., 2n, is a basis of R . Then we ∈ { } associate to z the complex valued coordinates zi = xi + iyi+n. Therefore, each point on R2n can be interpreted as a point of an n-dimensional vector space over the complex numbers, which we denote by Cn (a vector space is determined up to isomorphism by its dimension n). Obviously, the space Cn then is isomorphic to R2n [6,5].

2.3.2 A complex structure J on R2n

Let us deﬁne on Rm a real linear transformation J : Rm Rm from Rm to itself −→ such that

2 J = Idm, (2.17) − m where Idm is the identity mapping of R . Then J is called a complex structure on Rm. Also we can equip Rm with the structure of a complex vector space in the following manner:

(α + iβ)z := αz + βJz, α, β R. (2.18) ∈ Thus, scalar multiplication on Rm by complex numbers is provided and it is easy to convince yourself that Rm becomes a complex vector space. We call this process a complexiﬁcation of Rm. Suppose that the matrix of the complex j j structure J with respect to the basis ei , i = 0, ..., m is J = (a ), where a are real { } i i numbers. It is obvious that the eigenvalues of J are i, and they must appear in ± pairs. Therefore, the dimension of Rm must be even, i.e., m = 2n [11]. All this is indeed very similar to the previous identiﬁcation of R2n with Cn which we have performed by introduction of complex coordinates on R2n. We see now that it can also be done by using complex structure tensor J. To make this fact more clear we consider the converse direction by identifying Cn with 2n n R = x , ..., xn, y , ..., yn , xj, yj R. Scalar multiplication by i in C induces a { 1 1 } ∈ mapping

2n 2n J : R R given by J(x , ..., xn, y , ..., yn) = ( y , ..., yn, x , ..., xn), −→ 1 1 − 1 − 1 2 and, moreover, J = Id2n. The complex structure J deﬁned in this way is called − the standard complex structure on R2n [12, 11,6, 13,5].

2.3.3 Subgroup GL(n, C) of GL(2n, R) as a group preserving the complex structure J

As it was mentioned above, we can always consider the real space R2n equipped with a complex structure as the complex vector space Cn. The non-singular linear 2.3. COMPLEX STRUCTURE ON R2N 15 transformations of the vector space Cn map Cn to Cn and consequently preserve the complex structure. Moreover, they form a group, called the general linear group of degree n over C, denoted by GL(n, C). This group is isomorphic to the group of n n invertible matrices with complex entries. Upon realization of Cn, × each of these matrices corresponds to a linear transformation of the real vector space R2n Cn. That is why the group GL(n, C) can be regarded as a subgroup of GL(2n, R). The coset space GL(2n, R)/GL(n, C) determines all complex structures on R2n by the mapping [A] A 1 JA, where [A] is the equivalence class of A 7−→ − ∈ GL(2n, R) and J the standard complex structure on R2n [6, 11,5].

2.3.4 Complex structure J on the space R2p,2q and Hermitian metric g of signature (p, q)

Suppose we have the real vector space R2n, equipped with a complex structure J. Complexiﬁcation of R2n makes it possible to regard this space as the complex vector space Cn. In the same way, as before for a real space, we can deﬁne on Cn n n a symmetric sesquilinear form < , >C: C C C, which satisfy · · × −→

< ax + y, z >C = a < x, z >C + < y, z >C, (2.19)

< x, ay + z >C = < x, y >C +a¯ < x, z >C, (2.20)

< x, y >C = < y, x >C, (2.21) where we denote by x¯ the complex conjugate of x. We call the scalar product n < , >C the complex scalar product on C . · · In general, we can write the complex scalar product in the form expressed by the corresponding metric h of signature (p, q) (of course, p + q = n):

i j n < x, y >C= hijx y¯ , x, y C . (2.22) ∀ ∈ For an appropriate basis of Cp,q it can be transformed to

Xp Xn i j i i j j n < x, y >C= hijx y¯ = x y¯ x y¯ , x, y C , (2.23) − ∀ ∈ i=1 j=p+1

Here we see the connection with the metric h. The scalar product < , >C is · · called Hermitian scalar product or Hermitian form. In general, the matrix (hij) can take complex values. However, there is always a basis with respect to which the metric h is diagonal and real. Since the pseudo-Euclidean metric h has the signature (p, q), it is more convenient to write R2p,2q for the space R2n and Cp,q for Cn. If we now consider the Hermitian form h on the complex vector space Cn, understood as a complexification of R2n by J, we will find out that the ‘realized’ 16 CHAPTER 2. MATHEMATICAL PRELIMINARIES form of h, i.e. its corresponding form on R2n, can be given by a positive definite real bilinear form g on R2n, i.e., by

2n h(ξ , ξ2) = g(ξ , ξ2) + ig(Jξ , ξ2), ξ , ξ2 R . (2.24) 1 1 1 ∀ 1 ∈ A special feature of the pseudo-Euclidean metric g is that it satisﬁes the compatibility condition with the complex structure J in the sense that

2n g(Jξ , Jξ2) = g(ξ , ξ2), ξ , ξ2 R . (2.25) 1 1 ∀ 1 ∈ In this fashion one says that g is J invariant and also calls it Hermitian metric or J-metric on R2n. We note that (pseudo-)Hermitian metrics h on Cp,q and J- invariant (pseudo-)Euclidean metrics g stand in one to one correspondence to each other [6,5,8].

2.3.5 The group U(p, q) preserving J and g Let R2p,2q be equipped with a complex structure J and a pseudo-Hermitian metric g of type (p, q). As the unitary group U(p, q) we deﬁne a group of linear transformations on R2p,2q Cp,q, which preserve the complex scalar product deﬁned in previous section. In other words, if

< Ax, Ay >C=< x, y >C, (2.26) then A U(p, q). In fact, U(p, q) consists of those complex n n matrices A which ∈ × are elements of GL(n, C) and satisfy

T A¯ gA = g = (gij), (2.27) because

T T T T < Ax, Ay >C= (Ax) g(Ay) = x A gA¯y¯ = x gy¯. (2.28)

The last equation implies

¯T A Idp,qA = Idp,q, (2.29) if we put gij = (Idp,q)ij. Consequently we can identify U(p, q) with a group T 1 consisting of the matrices A, which obey A† := A¯ = Idp,qA− Idp,q. We now characterize the (pseudo-)unitary group U(p, q) as intersection of the groups GL(n, C) and SO(2p, 2q). Obviously, U(p, q) as a set of non-degenerate transformations of the complex space is a subset of GL(n, C). Further we intro- 2p,2q duce on R the usual pseudo-Euclidean scalar product < , >R, deﬁned for a 1 1 1 n 1 1 1 · · n pair of vectors ξ1 = (x1, ..., x1, y1, ..., y1) and ξ2 = (x2, ..., x2, y2, ..., y2) as Xp Xn i i i i j j j j < ξ , ξ2 >R = (x x + y y ) (x x + y y ). (2.30) 1 1 2 1 2 − 1 2 1 2 i=1 j=p+1 2.3. COMPLEX STRUCTURE ON R2N 17

Then the complex scalar product is related to this one in the following way:

Re < x, y >C = < x, y >R, (2.31) where Re stands for a real part of a complex number. The explicit computation conﬁrms this equation,

Xp Xn i i i i j j j j Re < ξ , ξ2 >C = (x + iy )(x iy ) (x + iy (x iy )) 1 1 1 2 − 2 − 1 1 2 − 2 i=1 j=p+1 p n X X j j j j = (xi xi + yi yi ) (x x + y y ). (2.32) 1 2 1 2 − 1 2 1 2 i=1 j=p+1

Since < ξ, ξ >C is real, it follows

< ξ, ξ >C=< ξ, ξ >R . (2.33)

We conclude that a linear transformation A GL(n, C) preserves the complex ∈ scalar product < ξ, ξ >C if and only if A preserves < ξ, ξ >R. We infer that indeed A has to be an element of SO(2p, 2q), because SO(2p, 2q) is exactly the group leaving the form < ξ, ξ >R invariant (for the fact that the determinant of realized form of A is 1 look [6]). Hence, it follows that

U(p, q) = GL(2n, C) SO(2p, 2q). (2.34) ∩ 2.3.6 Euclidean case as a subcase

In the case when in the above considerations the metric g is simply the Hermitian metric, i.e., g is of type (n, 0), the unitary group U(p, q) with p = n, q = 0 turns out to be the intersection of the general linear n-dimensional group on C and the usual special orthogonal group of 2n dimension:

U(n) = GL(n, C) SO(2n). (2.35) ∩

2.3.7 Symplectic structure ω on R2n and R2p,2q Suppose we have endowed the space R2n with a complex structure J and a Hermitian metric g. Since g is J-invariant, one can deﬁne a symplectic form ω on R2n by

2n ω(ξ , ξ2) = g(Jξ , ξ2), ξ , ξ2 R . (2.36) 1 1 ∀ 1 ∈ The form ω is in fact antisymmetric, since x, y R2n ∀ ∈ ω(x, y) = g(Jx, y) = g(J2x, Jy) = g(x, Jy) = g(Jy, x) = ω(y, x). (2.37) − − − 18 CHAPTER 2. MATHEMATICAL PRELIMINARIES

Furthermore, ω is non-degenerate, because g is non-singular. Moreover, we observe that ω is invariant under the action of J:

ω(Jx, Jy) = g(J2x, Jy) = g(J3x, J2 y) = g(Jx, y) = ω(x, y). (2.38)

The two-form ω is called the Kahler¨ form. In components we can write it as i j k i j k ωijx y = gkj Ji x y or ωij = Ji gkj. That implies that in matrix form: ω = Jg [14].

2.3.8 U(p, q) as a subgroup of Sp(2n, R) We have seen that the unitary group is the group which preserves the standard complex structure J as well as any pseudo-Hermitian metric g of signature (p, q). Since the Kahler¨ form ω is constructed from g and J, it follows automatically that the symplectic group, deﬁned as a group preserving a symplectic form ω, contains linear transformation A U(p, q). Hence, we conclude that ∈ U(p, q) Sp(2n, R). (2.39) ⊂ 2.4 Algebraic structures on vector spaces

Now let us take a look at vector spaces from the algebraic side. We will conﬁrm ourself that group theoretical properties stand in an non-separable conjunction with algebraic structures.

2.4.1 General notion of algebra

Before proceeding further we introduce a general notion of algebra. To do that we need a definition of objects such as field and vector space. In mathematical sense a field K is a set equipped with two binary operations, + : K K K × −→ and : K K K, called addition and multiplication, respectively. The result · × −→ of applying them to elements a, b K is denoted by a + b and ab. The set K ∈ additionally must satisfy the following requirements known as field axioms. Addition:

Commutativity: a + b = b + a; • Associativity: a + (b + c) = (a + b) + c; • Existence of negative: for any a K there is an element a in K, which • ∈ − obeys a + ( a) = 0 (written also as a a = 0); − − Existence of zero: there exists an element 0 K such that a + 0 = a, a K; • ∈ ∀ ∈ Multiplication:

Commutativity: ab = ba; • 2.4. ALGEBRAIC STRUCTURES ON VECTOR SPACES 19

Associativity: a(bc) = (ab)c; • Existence of unity: a K there exists an element 1 K with the property • ∀ ∈ ∈ 1a = a1 = a;

1 Existence of inverse: for any a K there exists an element a− such that • 1 1 ∈ aa− = a− a = 1 (it can be shown that this element is unique);

Addition and multiplication:

Distributivity: (a + b)c = ac + bc and c(a + b) = ca + cb. • Moreover, we exclude the trivial case when K contains only zero element. Now we go over to the notion of a vector space. Under a vector space over a field K we understand a set V together with two binary operations V V V × −→ and K V V, called vector addition and scalar multiplication. The elements × −→ of V are called vectors. The sum of two vectors is denoted by v + w, the product of a scalar a K and a vector v is denoted a v or simply av. ∈ · To qualify the space as a vector space, addition and multiplication have to fulfill a number of axioms described below. Let u, v, w V and a, b K. First of ∈ ∈ all, for a vector space the addition field axioms must hold (see above). Further, a vector space is required to satisfy some axioms for the multiplication on scalars:

a(bv) = (ab)v; • (a + b)v = av + bv; • a(v + w) = av + aw; • 1v = v. • Now we have all ingredients to give the deﬁnition of an algebra. Suppose K is a ﬁeld and V is a vector space over K equipped with an additional binary operation from V V to V, denoted here by and called multiplication on V. × · Then A = (V, ) is an algebra over K or a K-algebra if the following identities hold · for any three elements v, w, and u in A, and all elements a and b in K:

Left distributivity: (x + y)z = xz + yz; • Right distributivity: x(y + z) = xy + xz; • Compatibility with multiplication on scalars: (ax)(by) = (ab)(xy). • These three axioms are equivalent to the statement that the binary operation is bilinear. We note that multiplication of elements of an algebra is not necessarily associative. We discuss in the following two possible kinds of algebra: associative and non-associative one [5, 15]. 20 CHAPTER 2. MATHEMATICAL PRELIMINARIES

2.4.2 Associative and non-associative algebras

An associative algebra A is a K-algebra, for which the associativity law holds:

v(wu) = (vw)u = vwu, v, w, u A. (2.40) ∀ ∈ If A contains an identity element, i.e., an element 1 such that 1v = v1 = v for all x in A, then we call A an associative algebra with a unit. In this case all elements a of the field K identified with 1a are in A. As examples of associative algebras we mention the algebra of n n matrices, the algebra of polynomials with real × coefficients and Clifford algebras. Under a non-associative algebra we understand an algebra over a field K, for which the associativity law does not hold. In this case we distinguish the right multiplication Ra : A A A, v va and the left multiplication La : × −→ 7−→ A A A, v av. We call an algebra unital or unitary if it has a unit or × −→ 7−→ identity element I with Iv = v = vI for all v in the algebra. Well known examples of non-associative algebras are Lie algebras, hyperbolic quaternion algebra (the mathematical concept was first suggested by A. Macfarlane), Poisson algebras and many others [5].

2.4.3 Algebra of quaternions H

A special kind of associative algebra is the quaternion algebra. At ﬁrst we introduce the notion of quaternions. The set of quaternions H is a four dimensional vector space over real numbers consisting of all linear combinations

q H, q = a + bi + cj + dk, (2.41) ∈ where a, b, c, d are real numbers and 1, i, j, k are linear independent vectors. Mul- tiplication among the vectors i, j, k is established as

ij = k = ji, jk = i = kj, ki = j = ik, i2 = j2 = k2 = 1. (2.42) − − − − Thus, the vectors i, j, k have the property of imaginary unities. We can extend the deﬁnition of multiplication to all elements of H if use the distributive laws and assume that the unity 1 commutes with all quaternions (then for example 1 = − ijk). Therefore, with this multiplication and with the usual linear addition H becomes an associative non-commutative algebra over the ﬁeld of real numbers [6,5].

2.4.4 The group Sp(1) SU(2) as a group preserving the quaternion structure

Note that the imaginary units of the quaternion algebra admit a representation in the algebra of 2 2-matrices, where the real unit vector is the diagonal unit × 2.4. ALGEBRAIC STRUCTURES ON VECTOR SPACES 21

matrix Id2. If we construct out of two matrices ! ! a b d e A = and B = , (2.43) c a f d − − with arbitrary components a, b, c, d, e, f and vanishing trace, two imaginary units

A 2 B 2 j = = j = Id2, k = = k = Id2, √det A ⇒ − √det B ⇒ − then the product of them ! AB 1 ad + b f ae bd jk = = − (2.44) √ √ cd a f ec + ad det AB det AB − is again an imaginary unit under the condition that the product matrix is trace- less: 2ad + b f + ec = 0. If we denote jk = l, then it is straightforward to prove that the matrices I = Id2, j, k, l obey the multiplication rules of the quaternion algebra. An example of such imaginary triple j, k, l is well known Pauli matrices (with factor complex factor i) appearing in the quantum mechanics: − ! ! ! 0 1 0 i 1 0 j = i , k = i − , l = i . (2.45) − 1 0 − i 0 − 0 1 − This correspondence leads to the realization of (2.42) in terms of 2 2 matrices. We × then say that Pauli matrices are a 2 2 complex realization of abstract quaternion × algebra. We note that there are other possibilities to represent imaginary units of the quaternion algebra. We mention only that the quaternion basis unit vectors can also be written using real 4 4-matrices. × Suppose, we choose an appropriate representation of the quaternion algebra. Then one can rise the question which transformation of the units does not change the quaternion multiplication rules. To give an answer it is convenient to change the notation for quaternion algebra and to rewrite the rules in the compact form:

1qi = qi1 = qi, qiqj = δij + q , (2.46) − ijk k where q1,2,3 = j, k, l and ijk is the antisymmetric Levi-Civita tensor in three dimensions. So let U be an arbitrary transformation. Let U also be non-zero element of 1 some set G with unity Id (identity transformation), so that there exists U− with

1 1 UU− = U− U = Id.

1 Since U− is also an element of G, the last relation assumes that a multiplication in G is provided and thus requires G to be a group. Then we can impose the conditions

1 1 qk0 = UqkU− , 10 = U1U− = Id1 = 1, (2.47) 22 CHAPTER 2. MATHEMATICAL PRELIMINARIES leaving the multiplication rules (2.46) invariant

1 1 1 10qk0 = 1UqkU− = Uqk1U− = UqkU− = qk0 , (2.48) 1 1 1 qi0 qj0 = UqiU− UqjU− = UqiqjU− (2.49) 1 1 = UδijU− + Uq U− = δij + q . (2.50) − ijk k − ijk k0 It should be noted that any transformation U acts on the real unit as the identity transformation, so that in fact only imaginary units are involved. That is easy to understand, because in any matrix representation there exists a unique real unit - the unit diagonal matrix - which is preserved by action of an invertible matrix U. For this reason the transformation U is actually a transformation of the three vector units and does not concern the scalar part of a quaternion in any way. The number of free parameters of the transformation can be easily counted if we use the 2 2 matrix representation of the algebra. In this case the transformation × U is an unimodal 2 2 matrix, det U = 1, because the transformations (2.48) × do not depend on the determinant of U. Therefore only three components of U are independent. In general case when the parameters are complex, the total number of parameters is 6 and the group G is the group SL(2, C). In the case when these parameters are real, the number of free components is 3 and the group contracts to the group SU(2). In particular, since det U = 1, the transformation U in the 2 2-matrix repre- × sentation can be expressed as an unit quaternion, that is, if the Euclidean norm q 2 = qq¯ = a2 + b2 + c2 + d2 for q = a + bj + ck + dl is 1. Then any linear invertible | | transformation of the space H preserving the quaternion structure can be written in the form

1 αp : q pqp , 7−→ − where p H with p 2 = 1 and p 1 = p¯/ p 2. An alternative method to regard ∈ | | − | | transformation of quaternions is to write them as vectors and consider their transformations given by 3 3-matrices O: ×

qk0 = Ok0kqk. (2.51)

We remark only that the transformation group then is the group SO(3, R) (we refer the reader to [16,5]). Another way to treat the quaternion transformations is expressing quaternions q as

q = a + bj + ck + dl = (a + jb) + (c + jd)k = x + yk, (2.52) where x = a + jb, y = c + jd are complex numbers. Then q can be considered as a vector in C2. Moreover, we can deﬁne (analogous to the Hermitian form) on 2.4. ALGEBRAIC STRUCTURES ON VECTOR SPACES 23

pairs of vectors q1, q2 in H a form

< q , q2 >H= q q¯2 = (x + y k)(x2 + y2k) = (x x¯2 + y y¯2) + (y x2 x y2)k. (2.53) 1 1 1 1 1 1 1 − 1 We denote by Sp(1) the group of transformation of H which preserves the form (2.53). It follows that Sp(1) consists of transformations leaving the Hermitian form x1x¯2 + y1 y¯2 invariant and also preserving the skew-symmetric form y1x2 2 − x1 y2. Hence, Sp(1) contains those unitary transformations of C (i.e., elements of U(2)) which preserve the above skew-symmetric form. So we have shown that in fact Sp(1) is a subgroup of U(2). Moreover, we prove that Sp(1) is isomorphic to SU(2) in the following way. We consider the form y x2 x y2 as the area of a 1 − 1 complex parallelogram in C spanned by the side vectors (x1, y1) and (x2, y2). We conclude that Sp(1) is a group of unitary matrices with determinant 1, since they do not change the orientation of the parallelogram [6].

2.4.5 Vector space Hn and the group Sp(n) We have seen that the space H can be expressed in terms of the complex space C2. In a simple way we can expand our construction on a multidimensional quaternion space Hn. For that purpose we write again q H as ∈ 1 1 q = a + bi + cj + dk = (a + ib) + (c + id)j = x e1 + y (je1), (2.54) so that we can now identify the space Hn with the complex space C2n equipped 1 n 1 n with the basis e1, ..., en, je1, ..., jen and with complex coordinates x , ..., x , y , ..., y . Then for each q Hn we can write q = qke = (xke + yk je ). ∈ k k k We denote by GL(n, H) the group of linear invertible transformation on Hn and consider again the group Sp(n) GL(n, H) preserving the form < , >H, ⊂ · · which in n dimensions has the form Xn Xn i i i i i i < q1, q2 >H = q1q¯2 = (x1 + y1 j)(x2 + y2 j) i=1 i=1 Xn Xn = (xi x¯i + yi y¯i ) + (yi xi xi yi )j, (2.55) 1 2 1 2 1 2 − 1 2 i=1 i=1 Again as before it follows that the group Sp(n) consists of transformations which n C2n P i i i i leave the Hermitian form on and the skew-symmetric form (y1x2 x1 y2)j i=1 − invariant. This implies that Sp(n) is a subgroup of U(2n) preserving this skew- symmetric form [6].

2.4.6 Metric of signature (p, q) with p + q = n on Hn and the group Sp(p, q) Analogously to the preceding steps we can extend the concept deﬁned in the previous section to an arbitrary metric g with signature (p, q) (with p + q = n) on 24 CHAPTER 2. MATHEMATICAL PRELIMINARIES

Hn,

i j < q1, q2 >H= gijq1q¯2, (2.56) which for an appropriate basis vectors takes the form

Xp Xn i i i i < q , q2 >H = q q¯ q q¯ (2.57) 1 1 2 − 1 2 i=1 i=p+1 Xp Xn = (xi x¯i + yi y¯i ) (xi x¯i + yi y¯i ) 1 2 1 2 − 1 2 1 2 i=1 i=p+1 Xn Xn + (yi xi xi yi )j (yi xi xi yi )j. (2.58) 1 2 − 1 2 − 1 2 − 1 2 i=1 i=p+1

We see that it is quite obvious to identify the group Sp(p,q) preserving the above form of signature (p, q) as the subgroup of U(p,q), which leaves the metric g in n n P i i i i P i i i i variant and preserves the antisymmetric form (y1x2 x1 y2)j (y1x2 x1 y2)j i=1 − −i=p+1 − of type (p, q).

2.4.7 Algebra O of octonions

Graves’s octonions form the last one after Hamilton’s quaternions in a sequence of four increasingly complicated normed division algebras (i.e., algebras with well deﬁned division operation and norm obeying xy = x y ): the real k k k k · k k numbers R, the complex numbers C, the quaternions H and the octonions O. The octonions O forming a vector space can be formally expressed as 8- tuples (or as a octave) of real numbers. Every octonion can be written as a linear combination of unit octonions 1, e , ..., e7 : { 1 } 7 0 P i 0 7 O = a + a ei : a , ..., a R . { i=1 ∈ }

Defined in this way octonions constitute an algebra over the reals generated by units 1, e1, ..., e7 required to satisfy usual addition and certain multiplication laws. Addition of octonions is accomplished by adding corresponding coefficients, as it is for quaternions and complex numbers. Multiplication properties of octonions one can find in the Table 1.1. Analyzing commutativity and associativity we observe that the multiplication table is not commutative and is not associative. Furthermore, we note that the table does not seem to have any pattern. However, we can generate an equivalent table by swapping rows and columns as follows: we swap e7 and e2 then e4 and e6 then e6 and e7. The result is presented in Table 1.2, where we use another basis i, j, k, l, li, lj, lk. In this table we can see 2.4. ALGEBRAIC STRUCTURES ON VECTOR SPACES 25

ei ej 1 e e2 e3 e e5 e6 e7 · 1 4 1 1 e1 e2 e3 e4 e5 e6 e7 e e -1 e e7 e2 e6 e5 e3 1 1 4 − − − e2 e2 e -1 e5 e e3 e7 e6 − 4 1 − − e3 e3 e7 e5 -1 e6 e2 e e − − − 4 1 e e e2 e e6 -1 e7 e3 e5 4 4 − 1 − − e5 e5 e6 e3 e2 e7 -1 e e − − − 1 4 e6 e6 e5 e7 e e3 e -1 e2 − 4 − − 1 e7 e7 e3 e6 e e5 e e2 1 − 1 − 4 − − Table 2.1: Multiplication table for octonions [5]

a b 1 i j k l li lj lk · 1 1 i j k l il jl kl i i -1 k j li l lk lj − − − j j k -1 i lj lk l li − − − k k j i -1 lk lj li l − − − l l li lj lk -1 i j k − − − li il l lk lj i -1 k j − − − lj jl lk l li j k -1 i − − − lk kl lj li l k j i 1 − − − − Table 2.2: Multiplication table for octonions [5] that octonions can be thought as pairs of quaternions, i.e., they can in analogy with quaternions be written as

Y = a0 + a1i + a2 j + a3k + a4l + a5li + a6lj + a7lk (2.59) = (a0 + a1i + a2 j + a3k) + l(a4 + a5i + a6 j + a7k)

= q1 + lq2, (2.60) where q , q2 H. In fact, it can be done via Cayley-Dickson construction which 1 ∈ defines an octonion as a system of two quaternions, q = (a, b), with addition defined pairwise and multiplication defined as

(a, b)(c, d) = (ac db¯ , da + bc¯), (2.61) − where (a, b), (c, d) are pairs of quaternions and a¯ denotes the conjugate of the quaternion a. This deﬁnition is equivalent to the one given above when the eight unit octonions are identiﬁed with the pairs

(1, 0), (i, 0), (j, 0), (k, 0), (0, 1), (0, i), (0, j), (0, k). (2.62)

For more information we refer the reader to [17, 18,5, 19]. 26 CHAPTER 2. MATHEMATICAL PRELIMINARIES

2.4.8 Properties of the octonion structure constants

As it was shown above multiplication table for octonions determines relations between the single unit octonions entirely. Nevertheless, there is another way to express the multiplication rules between them. This is because octonions algebra has the same structure as the algebra of quaternions (compare (2.46)) and can be written in a similar short form:

eiej = δij + f e , eie0 = e0ei = ei, e0e0 = 1, (2.63) − ijk k where i, j, k run from 1 to 7 and fijk are the structure constants. Using the multiplication table it can be proved that the structure constants are completely antisymmetric in their indices. Also it can be shown that only for the combinations of indices [20]:

(123), (145), (176), (246), (257), (347), (365), (2.64) the structure constants do not vanish and are equal to one.

2.4.9 Group G2 and Spin(7) related with the octonion algebra In many cases it can be useful to consider linear transformations of the vector space of octonions, so we want to ﬁnd inner symmetries of the space O and analyze transformation behavior of its elements. The group of invertible linear transformations A from O to O, satisfying A(xy) = A(x)A(y), is called the automorphism group of O. It is easy to show that automorphisms of O in fact form a group, denoted by Aut(O) and also denoted by G2. One can show that G2 is a simply connected compact simple real Lie group of dimension 14. It is a subgroup of the group Spin(7). Moreover, this group is the smallest exceptional Lie group [17,5, 21]. Furthermore, G2 can be understood as a simple, 14-dimensional stabilizer of the three-form

ω := dx127 + dx347 + dx567 + dx135 dx146 dx236 dx245 Λ3(R7). (2.65) − − − ∈ 123 1 2 3 Here dx = dx dx dx . Under the action of G2 familiar vector spaces split ∧ ∧

so(7) = g2 g⊥, (2.66) ⊕ Λ2(R7) = Λ2 Λ2, (2.67) 14 ⊕ 7 Λ3(R7) = Λ3 Λ3 Λ3 . (2.68) 1 ⊕ 7 ⊕ 27 We will consider them later. Furthermore, it can be proven that the group G2 is the intersection of any two subgroups Spin(7) (of dimension 21) in the group Spin(8) (dim 28). The groups Spin(n) will be introduced in section 2.4.16[17, 21]. 2.4. ALGEBRAIC STRUCTURES ON VECTOR SPACES 27

2.4.10 Grassmann algebra: deﬁnition and main properties

Here, we wish to construct a space that is universal with respect to alternating multilinear maps. To do this, it is necessary at first to introduce the notion of tensor product of two vector spaces V, W over a field K. To construct V W, one begins with the set of ordered pairs in the Cartesian product V W = (v, w) v V, w W . Taking as a basis elements of the form × { | ∈ ∈ } (v, w) V W, we define the free vector space of V W: ∈ × × Pn F(V W) = αie(vi wi) n N, αi K, vi, wi V W , × {i=1 × | ∈ ∈ ∈ × } where the symbol e(v w) is used to emphasize that these are taken to be linearly × independent for distinct (v, w) V W. ∈ × Denoting by R the space generated by three equivalence relations

e((v +v ) w) e(v w) + e(v w), • 1 2 × ∼ 1× 2×

e(v (w +w )) e(v w ) + e(v w ), • × 1 2 ∼ × 1 × 2

ce(v w) e(cv w) e(v cw), • × ∼ × ∼ × where v, vi V, w, wi W and c K, the tensor product of the two vector spaces ∈ ∈ ∈ V and W is then the quotient space

V W = F(V W)/R. × We call the space V W the tensor product of V and W. Since in V W the space R is set to zero, vectors in V W obey the following relations:

(v1 + v2) w = v1 w + v2 w, •

v (w1 + w2) = v w1 + v w2, • c(v w) = (cv) w = v (cw). • This construction of the tensor product can be extended to the tensor product of more than two vector spaces. It arises in a natural way if we consider for example the case of three vector spaces V (W U) and (V W) U. It turns out that these spaces are equal up to an isomorphism. Then we say the tensor product is associative and write V W U. For this reason we deﬁne for an arbitrary natural number k the k-th tensor power of the vector space V as the k-fold tensor product of V with itself. That is

k k T (V):= V ... V = V (k-times tensor product). 28 CHAPTER 2. MATHEMATICAL PRELIMINARIES

Elements of Tk(V) are called tensors on V of rank k. By convention T0(V) is the ﬁeld K (as a one-dimensional vector space over itself). We then construct T(V) consisting of all possible Tk(V), k N. That is the direct sum of Tk(V) (under the ∈ direct sum of two vector spaces V, W over a ﬁeld K we understand the Cartesian product V W equipped with the following structure: (v , w2) + (v2, w2) = (v + × 1 1 v2, w + w2), a(v, w) = (av, aw), where vi V, wi W, a K; we denote it by 1 ∈ ∈ ∈ V W). We can write L T(V) = ∞ Tk(V) = K V V V V V V .... k=0 The multiplication in T(V) is determined by the canonical isomorphism

Tk(V) Tl(V) Tk+l(V) −→ given by the tensor product deﬁned in components as

(F G)i i ...in+m = Fi ...in Gin ...in+m , (2.69) 1 2 1 +1 where F is a tensor of rank n and G is a tensor of rank m. This multiplication rule implies that the tensor algebra T(V) is a naturally L graded algebra, i.e., it can be represented in the form A = Ai, where i runs i through Z, and multiplication satisﬁes the following condition: AiAj Ai+j. ⊂ Some Ai subspaces can be empty. The unit of algebra (if it exists) always belongs 2 to A0 (exterior algebra). Let A be the two-sided ideal generated by all elements of the form x x, x T(V). That is the set of all the elements of the form ∈ v1 ... vi ... vi ... vk where v V, k = 0, 1, 2, .... Then we deﬁne the space of all k-vectors, k N, ∈ ∈ called the exterior algebra or Grassmann algebra Λ(V) over a vector space V, as the quotient space

Λ(V):= T(V)/A, (2.70) i.e., the space T(V), where elements of A required to be set to zero. Since in Λ(V) we ignore the elements of A, it is convenient to introduce a new product called wedge product or exterior product given for a pair of v, w Λ(V) by ∈ v w = v w mod A. (2.71) ∧ It can be easily shown that wedge-product is antisymmetric. In particular, it obeys

x x ...x = sgn(σ)x ... x , (2.72) σ(1) ∧ σ(2) ∧ σ(k) 1 ∧ ∧ k 2 The term two-sided ideal is used in non-commutative rings (algebras) to denote a subset that is both a right ideal and a left ideal. In general, this is a subset I of a ring (or a algebra) R( , +), · for which (I, +) is a subgroup of (R, +) and xr, ry I, r R, x, y I. ∈ ∀ ∈ ∀ ∈ 2.4. ALGEBRAIC STRUCTURES ON VECTOR SPACES 29

where x , x2, ..., x V, σ is an element of the group of permutations of [1, ..., k] 1 k ∈ and sgn is the signature of the permutation σ. The space Λk(V) is the vector subspace of Λ(V) spanned by elements of the form

x ... x , xi V, i = 1, ..., k 1 ∧ ∧ k ∈ and is called an k-th exterior power of the space V. Elements of Λk(V) are said to be k-multivectors and can in general be written as X i1...ik 1 i1...ik A = A ei ... ei = A ei ... ei . (2.73) 1 ∧ ∧ k k! 1 ∧ ∧ k 1 i <...

(k + m)! ω η = Alt(ω η), (2.74) ∧ k!m! ⊗ where the alternation Alt of a multilinear map is given by

1 X Alt(ω)(x , ..., x ) = sgn(σ)ω(x , ..., x ). (2.75) 1 k k! σ(1) σ(k) σ S ∈ k Besides it is clear that any element of the exterior algebra can be written as a sum of k-multivectors. Hence, we obtain:

Λ(V) = Λ0 Λ1 Λ2 .... Thus, the dimension of the space Λ(V) is a sum of binomial coeﬃcients, which is equal to 2n [5,8, 13, 15, 22].

2.4.11 Clifford algebra of the vector space Rp,q of signature (p, q)

Clifford algebra constitutes an essential tool in the study of quadratic forms and have important applications in a variety of fields including geometry and theoretical physics. Quadratic forms are intimately connected with the theory of rotations in Rp,q generated by a the group Spin(p,q). The group Spin(p,q), called a spinor group, is defined as a certain subgroup of units of an algebra, Clp,q, the 30 CHAPTER 2. MATHEMATICAL PRELIMINARIES

Clifford algebra associated with Rp,q. Since the spinor groups are certain well- defined subgroups of units of Clifford algebras, it is necessary to investigate Clifford algebras to get a firm understanding of spinor groups. This section provides the main facts about Clifford algebra Cl(p, q) associated with a non- degenerate symmetric bilinear form of signature (p, q) and the corresponding group Spin(p,q). It also includes a study of the structure of Clifford algebras, which culminates in the 8-periodicity theorem of Elie Cartan and Raoul Bott [23]. Let Rp,q be a real vector space of dimension n, p + q = n together with a bilinear non-degenerate quadratic form φ : Rp,q Rp,q R of signature (p, q), × −→ and associated quadratic form Φ, v2 = Φ(v) = φ(v, v), v Rp,q. A Clifford algebra p,q p,q ∈ associated with R and Φ, denoted by Cl(R , Φ) = Clp,q(R), is a real associative algebra satisfying the following conditions [23, 24]:

1. Clp,q(R) is an associative real algebra with unit element e0,

p,q 2. There exists a map iΦ : R Clp q(R), obeying the fundamental identity: −→ , 2 p,q iΦ(v) = Φ(v) 1, v R . (2.76) · ∀ ∈ 3. For any real algebra A, and any linear map f : Rp,q A, with3 −→ ( f (v))2 = Φ(v) 1, v Rp,q, (2.77) · ∀ ∈

there is a unique algebra homomorphism, f¯ : Clp q(R) A, such that , −→

f = f¯ iΦ (2.78) · and the diagram below commutes

p,q iΦ R Clp q(R) −→ , f f¯ (2.79) &. A .

Since

Φ(u + v) Φ(u) Φ(v) = 2φ(u, v) (2.80) − − and

(i(u + v))2 = (i(u))2 + (i(v))2 + i(u)i(v) + i(v)i(u), (2.81) (i(v))2 = Φ(v) 1, (2.82) · 3 We warn the readers that for example [8] adopt the opposite to our sign convention in deﬁning Cliﬀord algebras, i.e., there is used the condition

( f (v))2 = Φ(v) 1, v Rp,q. − · ∀ ∈

The most confusing consequence of this is that then Clp,q(R) in [8] is our Clq,p(R). 2.4. ALGEBRAIC STRUCTURES ON VECTOR SPACES 31 we get

i(v)i(w) + i(w)i(v) = 2φ(v, w), v, w Rp,q. (2.83) ∀ ∈ p,q As a consequence, if (u1, ..., un) is an orthogonal basis of R (which means that ujuk = φ(uj, uk) = 0 for all j , k), we have [23,8]:

i(uj)i(u ) + i(u )i(uj) = 0, j , k. (2.84) k k ∀ Moreover, one can show that for each vector space equipped with a quadratic form there exists up to isomorphism an unique (Z2-graded) Cliﬀord algebra. We describe an approach how it can be constructed. Let us start with the most general algebra that contains Rp,q, namely the tensor algebra T(Rp,q), and then enforce the fundamental identity by taking a suitable quotient. Denote by I(Φ) the two-sided ideal of the full tensor algebra T(Rp,q) generated by elements of the form x x Φ(x) 1. Then a Cliﬀord algebra − · is given by

p,q Clp,q(R) = T(R )/I(Φ). (2.85)

p,q It is straightforward to show that Clp,q(R) contains R and satisﬁes the above commutation relations (2.84), so that Clp,q(R) is unique up to an isomorphism. Thus, one speaks of “the” Cliﬀord algebra of Rp,q [8,5, 23].

2.4.12 Generators of Clifford algebra

p,q n For a given basis e , ..., en of R , the 2 1 products 1 −

ei ei ...ei , 1 i < i2... < i n, 1 k n, 1 2 k ≤ 1 k ≤ ≤ ≤ and 1 form the Cliﬀord algebra Clp,q(R). The empty product (k = 0) is deﬁned as the multiplicative identity element. One can show that the dimension of Clp,q(R) n n is 2 unless e ...en is a scalar multiple of the identity, otherwise it is 2 1 [25, 23]. 1 − This exceptional case occurs for p q 1 divisible by 4[26]. Then if (e1, ..., en) is p,q − − an orthogonal basis of R , Clp,q(R) is the algebra presented by the generators (e1, ..., en) satisfying the relations

2 e = Φ(ej) 1, 1 j n, and (2.86) j · ≤ ≤ eje = e ej, 1 j, k n, j , k. (2.87) k − k ≤ ≤ Examples: Since in Cl0 0(R) there is no non-zero vectors it is easy to see that Cl0 0(R) R. , , ' Let q = n = 1, e = 1, and assume that Φ(x1e ) = x2. Then, Cl (R) is spanned 1 1 − 1 0,1 by the basis 1, e . We have e2 = 1. Under the bijection e i, the Cliﬀord { 1} 1 − 1 7−→ algebra Cl0,1(R), also denoted by Cl0,1(R), is isomorphic to the algebra of complex numbers C. 32 CHAPTER 2. MATHEMATICAL PRELIMINARIES

1 2 Now, let n = 2, e , e2 be the canonical basis, and assume that Φ(x e + x e2) = { 1 } 1 2 2 R (x1 + x2). Then, Cl0,2( ) is spanned by the basis 1, e1, e2, e1e2 . Furthermore, we − 2 2 2 4 { } have e2e = e e2, e = 1, e = 1, (e e2) = 1 . Under the bijection 1 − 1 1 − 2 − 1 −

e i, e2 j, e e2 k, 1 7−→ 7−→ 1 7−→ we can easily prove that quaternion identities hold, and thus the Cliﬀord algebra Cl0,2(R) is isomorphic to the algebra of quaternions H. 2 For Φ 0, it follows that e = 0, i = 1, ..., n, and the Cliﬀord algebra Cln(R) ≡ i ∀ is equivalent to the exterior algebra Λn(R)[5, 23].

2.4.13 The 8-fold periodicity of Clifford algebra

Surprisingly, but the three examples in the last paragraph are enough to provide us with a complete classification of Clp,q(R). It turns out that the real algebras Clp,q(R) can always be constructed as tensor products of the basic algebras R, C and H. Elie Cartan was the first who studied the real algebras Clp,q(R) as matrix algebras and discovered the 8-periodicity in 1908. After Cartan, the first proof was independently given by Raoul Bott in the 1960s, but after that many different alternative ways of showing this property have been given [5, 23]. To begin we recall that every non-degenerate quadratic over Rn has the form Φ(x) = x2 + ... + x2 x2 ... x2 after a suitable choice of basis. If p + q = n, then 1 p − p+1 − − n for simplicity here we write Clp,q for the Clifford algebra Clp,q(R, Φ) associated to this quadratic form Φ. Of particular importance are the Clifford algebras Cln,0 and Cl0,n. We shall now give a compact introduction to these algebras [27]. We use the notation R(n) (resp. C(n)) for the algebra Mat(R) of all n n real × matrices (resp. the algebra Mat(C) of all n n complex matrices). As we have × mentioned in the last section, it is not hard to show that

Cl0,1 = C, Cl1,0 = R R, Cl0,2 = H, (2.88) Cl2,0 = R(2) and Cl1,1 = R(2). (2.89)

Then one can prove the following isomorphisms (the reader can ﬁnd the proof in Gallier [23]):

Cl0,n+2 Cln,0 Cl0,2, Cln+2,0 Cl0,n Cl2,0, (2.90) Clp+1,q+1 Clp,q Cl1,1, (2.91) 4 n Here we extend the quadratic form on R to a quadratic form on Clp,q(R) by requiring that distinct elements of the form ei1 ...eik are orthogonal to one another whenever the eij ’s are orthogonal to each other. Since then generators eij anti-commute, we can transform the basis vectors in the form Φ(ei1 ...eik ) to standard order, such that

r 2 2 r Φ(ei ...ei ) = ei ...ei ei ...ei = ( 1) e ...e = ( 1) Φ(ei )...Φ(ei ), 1 k 1 k 1 k − i1 ik − 1 k where ( 1)r is an overall sign corresponding to the number of ﬂips needed to correctly order the − basis vectors (i.e., the signature of the ordering permutation). 2.4. ALGEBRAIC STRUCTURES ON VECTOR SPACES 33

p+q p q -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 − 0 R 1 C 2R 2 HR(2) R(2) 3 2HC(2) 2R(2) C(2) 4 H(2) H(2) R(4) R(4) H(2) 5 C(4) 2H(2) C(4) 2R(4) C(4) 2H(2) 6 R(8) H(4) H(4) R(8) R(8) H(4) H(4) 7 2R(8) C(8) 2H(4) C(8) 2R(8) C(8) 2H(4) C(8)

Table 2.3: Cliﬀord Algebras Clp,q, p + q < 8. for all n, p, q 0. Using these isomorphisms and the fact that the following ≥ propositions (see [23, 28]) hold,

R(m) R(n) R(mn) m, n 0, (2.92) ∀ ≥ R(n) R K K(n), where K = C or H, n 0, (2.93) ≥ C R C C C, (2.94) C R H C(2), (2.95) H R H R(4), (2.96) one can prove the periodicity theorem. This theorem, called after its discoverers also Cartan/Bott theorem, gives a complete classiﬁcation of real Cliﬀord algebras. It says for n 0 that ≥

Cl0,n+8 Cl0,n Cl0,8, (2.97) Cln+8,0 Cln,0 Cl8,0. (2.98) Cl0,8 = Cl8,0 = R(16). (2.99)

(for the proof see for instance [23, 27]). More general relations for Clp,q are:

Clp,q Clp 4,q+4, where p 4, (2.100) − ≥ Clp+8,q Mat(16, Clp,q), (2.101) where Mat(16, Cl(p, q)) is the algebra of 16-dimensional matrices with entries in Clp,q [8, 28, 29]. Based on the knowledge of these relations one can construct Table 2.3 of isomorphisms for every value of (p, q) (this table can be found in [5, 29]). Using Table 2.3 we can in particular write down the classification of the real Clifford algebras Cln,0 and Cl0,n. The Cartan/Bott theorem also tells us how to obtain the result for n > 8. The main point is that any real Clifford algebra Clr,s n is isomorphic to one of the matrix algebras K(2n) End(K2 ) or K(2n) K(2n), where K is either the reals R, the complex numbers C or the quaternionsH [28]. 34 CHAPTER 2. MATHEMATICAL PRELIMINARIES

Moreover, it turns out that from the classification for the Clifford algebras Cln,0 and C0,n one can actually obtain the classification for Clifford algebras over C. It is much simpler to provide the classification for Clifford algebras over C due to the fact that every non-degenerate quadratic form over Cn can be written C 2 2 as Φ (x) = x1 + ... + xn. For this reason to each dimension n there corresponds C n only one Clifford algebra Cln(C, Φ ) over C in the Clifford algebra. We denote C C Cln(C, Φ ) simply by Cln . p,q As to indefinite Clifford algebras Clp,q over R , we can complexify Clp,q by changing the real quadratic form by the form taken over complex numbers. We C then obtain Clp q C = Cl . Hence, using results on the classification of Clifford , ⊗ p+q algebras over R, we arrive at the classification for Clifford algebras over C. Thus, due to the periodicity theorem, we get for n 0: ≥ C C C C C Cl Cln Cl , Cl C C, Cl C(2). (2.102) n+2 2 1 2 The proofs can be found, e.g., in [23, 27].

2.4.14 Matrix representations of Clifford algebra over C and R

In order to get finished with Clifford algebras, we will need to use representations of the Clifford algebras over R and C. We start recalling the basics concerning representation theory. Let V be a vector space over K0 (K0 = R or C) and let Φ be a quadratic form on V. Let K K be a field containing K . Then a K-representation of Cl(V, Φ) is a ⊃ 0 0 K0-algebra homomorphism

ρ : Cl(V, Φ) HomK(W) −→ into the algebra of linear transformations of a finite dimensional vector space W over K. We call the vector space of representations W a Cl(V, Φ)-module5. We can also write φ w := ρ(φ)(w), (2.103) · where φ Cl(V, Φ), w W. The product φ w is referred to as Clifford multipli- ∈ ∈ · cation [27]. We call the map ρ a representation of Cl(V, Φ) in W. Furthermore, for many purposes it is important to know how the Cl(V, Φ)-module W decomposes, that is why we introduce the following definition. Namely, using the same notation as above, we say that the representation ρ is reducible if the vector space W can be written as a non-trivial direct sum W = W1 W2 such that φ Wi Wi, for · ⊂ i = 1, 2 and φ Cl(V, Φ). In this case we write the representation ρ as ρ = ρ1 ρ2, ∀ ∈ where ρi is a restriction of ρ to Wi. A representation is called irreducible if it

5According to [5] the abstract theory of Clifford modules was postulated by M.F.Atiyah, R.Bott and A.Shapiro. 2.4. ALGEBRAIC STRUCTURES ON VECTOR SPACES 35 is not reducible. One can show that any representation ρ of a Clifford algebra Cl(V, Φ) can be decomposed into a direct sum ρ = ρ1 ... ρr of irreducible representations (the proof can be found in [28, 27,5]). n Wenow concentrate on representations of the Clifford algebra Clp,q on W = R determining an (p + q)-dimensional subspace in End(Rn) R(n). Because of the classification in the previous sections, we can find all irreducible representations of Clifford algebras over R and C. Initially we will devote ourself to the construction of matrix representations of the Dirac matrices in Clp,q(C). We begin with the Clifford algebras of type Cp,0 where p is even. Similar to the construction of the Clifford algebra of quaternions Cl0,2(C) H we can use the Pauli spin matrices ! ! ! 0 1 0 i 1 0 σ = , σ = − , σ = (2.104) 1 1 0 2 i 0 3 0 1 − to find a matrix representation for Clp,0(C). Namely for Cl2,0(C), we can use γ = σ and γ2 = σ2. For C we set γ = σ σ , γ2 = σ σ2, γ3 = σ σ3 and 1 1 4,0 1 1 ⊗ 1 1 ⊗ 1 ⊗ γ = σ2 Id2. Here we use the tensor product, also named Kronecker product, 4 ⊗ of matrices. Via induction this construction can be continued for all Clp,0(C). If a matrix representation of size 2m 2m for C2m 0 is provided, we write for the × , representations of the basis elements simplified γk(2m), where k = 1, ..., 2m. Then a matrix representation for the Dirac matrices of Cl2m+2,0(C) can be expressed as: γ (2m + 2) = σ γ (2m), for k = 1, ..., 2m, (2.105) k 1 ⊗ k m γ m (2m + 2) = i σ J(2m), γ2m+2(2m + 2) = σ2 Id2m, (2.106) 2 +1 1 ⊗ ⊗ m where J(2m) = γ1(2m)...γ2m(2m)[25].Note that the factor i , which appears in the 2 second equation, guarantees that γ2m+1(2m + 2) = +Id2m+2. One can easily convince himself that the system of matrices γk(2m + 2) is 2 { } orthonormal and obeys γ (2m + 2) = +Id2m+2 k = 1, ..., 2m + 2. That is why it k ∀ generates the Clifford algebra Cl2m+2,0(C). After all that results it turns out that a construction of the Dirac matrices for Clifford algebras of the type Clp,q(C), where p + q = 2m, becomes much easier. All what one should do is to take the Dirac matrices constructed for Cl2m,0(C) and simply leave the first p matrices unmodified multiplying the rest q of them by i. Obviously, for p + q = 2m the elements of Clp,q(C) form a vector space of dimension 22m. Simultaneously, over C the set of complex 2m 2m-matrices × also form a vector space of dimension 22m. Hence, we see that for p + q = 2m, the Clifford algebra Clp,q(C) is isomorphic to C(2m), the algebra of complex 2m 2m-matrices. Consequently, all complex universal Clifford algebras for × non-degenerate pseudo-Euclidean spaces of a given dimension are equivalent. In the case if the dimension of the pseudo-Euclidean space is odd, it is not 2 much harder to show that Clp,q(C) is isomorphic to C(2m) = C(2m) C(2m). The 36 CHAPTER 2. MATHEMATICAL PRELIMINARIES corresponding Dirac matrices can be constructed iteratively by

γ (2m + 1) = σ3 γ (2m), for k = 1, ..., 2m, (2.107) k ⊗ k m γ m (2m + 1) = ( i) σ3 J(2m), (2.108) 2 +1 − ⊗ (see Problem 11.1 in [25]). The situation of real Clifford algebras in general is more involved. It is not trivial any more to find an appropriate matrix representation for it. As we have already seen in Table 2.3, all real Clifford algebras are isomorphic to a matrix algebra of the form K(2n) or K(2n) K(2n), where K either R, C ⊕ or H. The classification of the real Clifford algebras is a little more intricate. Nevertheless, knowing a representation of the first three real algebras, more precisely, of Cl1,1(R), Cl2,0(R) and Cl0,2(R), enable oneself to construct all real representations of Clp,q(R) for q , 0. Such a construction one can find in the reference [28]. In fact, it leads directly to isomorphisms

Cl1,1(R) Clp,q(R) Clp+1,q+1(R), (2.109) Cl0,2(R) Clp,q(R) Clq,p+2(R), (2.110) Cl2,0(R) Clp,q(R) Clq+2,p(R), (2.111) which allow to get representations of a large amount of Clifford algebras. An explicit construction of the rest of real representations can be proceeded by using the classification Table 2.3. Another strategy of the construction of a real representation of real Clifford algebras is also given in the paper of Okubo [30]. Some further information about real representations of Clifford algebras the reader can also find in [28, 27].

2.4.15 Spinors (fermions) as representation spaces of Clifford algebras

A spinor module S for the Clifford algebra Cl2k(C) is given by a choice of a 2k- dimensional complex vector space S, together with an identification Cl2k(C) = End(S) of the Clifford algebra with the algebra of linear endomorphisms of S. So we can understand a spinor space as a complex vector space S, together with an instruction of how the 2k generators ei of the Clifford algebra act as linear transformations on S. In order to construct such a spinor module S, together with appropriate operators on it, we can make use of exterior algebra techniques. Considering n n the real exterior algebra Λ∗(R ), we assign to a vector v Λ∗(R ) an operator on n ∈ Λ∗(R ) given by exterior multiplication by v:

v : a v a. ∧ −→ ∧ 2.4. ALGEBRAIC STRUCTURES ON VECTOR SPACES 37

n n A chosen inner product on R always induces inner product on Λ∗(R ), the one such that the wedge products of orthonormal basis vectors are orthonormal. Hence, with respect to this inner product there exists an adjoint operator to the operator v , which we denote by ι(v). So that < v a, b >=< a, ι(v)b >. ∧ ∧ We use here a physical notation for these operation:

a†(v) = v , a(v) = ι(v) and (2.112) ∧ a†(ei) = ai†, a(ei) = ai. (2.113) The algebra of these operation is known as the algebra of Canonical Anticom- mutation Relations, since the 2n operators ai† and ai satisfy

ai, aj = 0, a , a = 0 and ai, a† = δij. (2.114) { } { i† †j } { j }

Here we see the connection with the fermions: the ai† and ai turn out to be the creation and annihilation operators. n By construction this algebra can be represented as operators on Λ∗(R ). Fur- ther, we can identify this algebra with the Cliﬀord algebra Cln(R) as follows

Cln(C) v a (v) a(v), 3 −→ † − where one can easily see that

2 2 2 2 2 v = (a†(v) a(v)) = (a†(v)) + (a(v)) a†(v), a(v) = v 1. (2.115) − − { } −|| || n For even dimension n = 2k we can complexify Λ∗(R ) to get a complex representation of Cl2k(C) on this space. However, this representation is of dimension 22k and not the 2k-dimensional irreducible representation S we are searching for. For our aim we choose a complex structure J on V = R2k. Then V C can be ⊗ decomposed as V C = WJ WJ, where WJ is the +i eigenspace of J, WJ is the ⊗ ⊕ i eigenspace. In particular, we can pick an orthogonal complex structure J, in − the sense that < Jv, Jw >=< v, w >. Moreover, we observe that the quadratic form coming from the inner product vanishes on WJ. Thus we conclude that the Cliﬀord algebra Cl(WJ) can be identiﬁed with the exterior algebra Λ∗(WJ). We remark that it has the right dimension 2k. 2k Choosing the standard complex structure on V = R , by setting wj = e2j 1 + − ie2j, j = 1, ..., n, we can identify

k Cl2k(C) = End(Λ∗(C )). (2.116)

In terms of the creation and annihilation operators ai†, aj the generator of the Cliﬀord algebra are given as follows

e2j 1 = a† aj, e2j = i(a† + aj), j = 1, ..., k. (2.117) − j − − j

One can prove that the Cliﬀord algebra relations hold: ei, ej = 2δij. { } − 38 CHAPTER 2. MATHEMATICAL PRELIMINARIES

In addition to this, the set of all spinors can be decomposed in two subsets k S . It can be seen by considering the eigenvectors in Λ∗(C ) of the operator 1± i 1 k e2j 1e2j = [aj, a†], which are . The subspaces of Λ∗(C ) consisting of vectors 2 − 2 j ± 2 which are simultaneously the eigenvectors of all e2j 1e2j are called the weights − 1 of the representation S. Each of the weights corresponds to a set of k values 2 or 1 1 1 :( , ..., ). The decomposition into half-spin representations S = S+ S − 2 ± 2 ± 2 ⊕ − then corresponds to the decomposition into weights with even or odd number of minus signs [5, 31, 32].

2.4.16 Group Spin(p, q) Let K be a field R or C and V a finite-dimensional vector space over K. Also let Φ be a quadratic form on V of signature (p, q), < x, y >= 1 (Φ(x + y) Φ(y x)) 4 − − the associated bilinear form. We assume that < , > is non-degenerate, i.e., · · < x, y0 >= 0, for all x V, implies y0 = 0. We denote by Cl(V) the Clifford ∈ algebra associated to V and Φ. Since the Clifford algebra Cl(V) = T(V)/A can be considered as a quotient space of T(V), the grading of the tensor algebra T(V) induce a grading on Cl(V) in a natural way:

M∞ M∞ T(V) = Tk(V) = Cl(V) = Clk(V), (2.118) ⇒ k=0 k=0 where Clk(V) denotes the subset of the elements γ of the order k. If γ Cl(V) and the degree deg γ = k, then we deﬁne the grading map ∈ α : Cl(V) Cl(V) as α(x) = ( 1)kx. The set Γ of all u in Cl, such that u has an −→1 − inverse u− and

uVα(u) 1 V, i.e., uxα(u) 1 V for all x V, − ∈ − ∈ ∈ is a group under multiplication, which is called the Cliﬀord group of Φ. If u 1 belongs to the Cliﬀord group Γ of Φ, then su : x uxα(u) is an orthogonal −→ − transformation, i.e., it preserves Φ, because

1 1 2 2 1 1 Φ(uxα(u)− ) 1 = (uxα(u)− ) = ux u− = u(Φ(x) 1)u− · ± ± · = Φ(x) 1. (2.119) ·

Hence, the correspondence χ : u su is a linear representation of Γ, which is 7−→ called the vector representation of Γ [33]. Extending the transpose map t on T(V), which assigns to each element x 1 ⊗ ... xk T(V) the element xk ... x1, to the transpose on Clp,q(V) we introduce the ⊗ ∈ ⊗ ⊗2 t square-norm on Clp q(V) via x = xx¯, where we set x¯ = α(x ). In these fashion , k k we deﬁne the pin group as:

2 Pinp q(V):= x Γp q(V): x = 1 (2.120) , { ∈ , k k ± } 2.4. ALGEBRAIC STRUCTURES ON VECTOR SPACES 39 and the spin group as a subgroup of it,

+ Spin (V):= Pinp q(V) Γ (V), (2.121) p,q , ∩ p,q + + + with Γ (V) = Γ Cl , where we denote Cl (V) = u Clp q(V) : degree of u is p,q ∩ p,q p,q { ∈ , even [23,8, 29, 26,5]. } 2.4.17 Supersymmetry and superalgebra

For a long time it was a goal of many physicists to obtain a uniﬁed description of all basic interactions of nature, i.e., strong, electroweak, and gravitational interactions. Several attempts have shown that a necessary ingredient in any unifying approach is supersymmetry (SUSY) (for a physical introduction to supersymmetry see for example [34, 35, 36]). The algebra involved in SUSY is a Z2-graded Lie algebra which is closed under a combination of commutation and anti-commutation relations. Under the Z2-grading the algebra elements decom- pose into bosons, the even elements, and the fermions, the odd elements of the algebra. Such an algebra is called a Lie superalgebra. Supersymmetry transforms bosons into fermions and vice versa. The generators Qα of these transformations should therefore have fermionic character, i.e., they are generated by objects that transform in the spinor representations [5]. In its general form a superalgebra is an Z2-graded algebra containing Poincare´ algebra

[P , P ] = 0, [P , M ] = η P η P , (2.122) µ ν ρ µν ρµ ν − ρν µ [M , M ] = (η M + η M η M η M ) (2.123) µν ρσ − µρ νσ νσ µρ − µσ νρ − νρ µσ as well as a internal symmetry generated by the algebra elements Tr:

[Tr, Ts] = frstTt. (2.124)

In particular the supersymmetry must be a direct sum of both algebras (this fact follows from the Coleman-Mandula theorem [37]). This is achieved by the requirement that

[Tr, Pµ] = 0 = [Tt, Mµν]. (2.125) In addition, the supersymmetry includes a fermionic structure which can be i given by the set of anticommuting generators Qα (i = 1, ..., N) of the Lorentz algebra. One can choose Qi belonging to the ( 1 , 0) (0, 1 ) representation, i.e., α 2 ⊕ 2

1 β [Qi , M ] = (σ ) Qi . (2.126) α µν 2 µν α β

The Z2-graduation means that the commutator of two even (bosonic) generators (Pµ, Mµν, Tr) has bosonic nature, the commutator of two odd (fermionic) 40 CHAPTER 2. MATHEMATICAL PRELIMINARIES

i generators Qα has bosonic character and the commutator of an odd and an even generators is again fermionic. This implies that

i i j i β j [Qα, Tr] = (lr)jQα + (tr)j(iγ5)αQβ, (2.127)

i i where the summands (lr)j + iγ5(tr)j deﬁne a representation of the Lie algebra of the inner symmetry, and

i [Qα, Pµ] = 0. (2.128)

These equations are consequences of the generalized Jacobi identities. The last remaining relation is the relation between two fermionic generators which in a SUSY algebra must have the form

i j ij µ ij ij [Qα, Qβ] = 2(γµC)αβδ P + CαβU + (γ5C)αβV , (2.129) where Uij and Vij are even antisymmetric generators which commute with all generators included with itself. We call them center charges. The matrix Cαβ is the charge conjugation matrix (see for example [37]). The supersymmetry algebra, in its simplest form (N = 1) involves the gener- 1 ators of the Poincare´ algebra together with a self-conjugate spin 2 generator Qα, α = 1, 2. Recall that the generator of the Poincare´ algebra in tensor representation are given by the space-time rotations Mµν and translations Pµ. The fermionic 1 generators Qα and Qα∗ can be expressed as the Weyl spinors in the (0, 2 ) and 1 ( 2 , 0) representations of the spinor group SL(2, C), respectively. The algebra that deﬁnes supersymmetry then is given by [5, 38, 35]

[Qα, Pµ] = [Qα∗ , Pµ] = 0, (2.130) µ Q , Q∗ = 2σ P , (2.131) { α β} αβ µ Q , Q = Q∗ , Q∗ = 0, (2.132) { α β} { α β} where the indices take the values α, β = 1, 2 and σµ denotes the Pauli matrices, and by the commutation relation of the Poincare´ group (2.122-2.123). From the analysis of this section, we see that the SUSY algebra with the supercharges Qα and Qα∗ is equivalent to a Cliﬀord algebra Cl2(C). (For supersymmetry in 4- dimensional Euclidean see [39].) This is the simplest supersymmetric extension of the Poincare´ group known as Wess-Zumino model. Another models of supersymmetric algebras such as superconformal algebra and (anti-)deSitter algebras one can ﬁnd in reference [37].

2.5 Theory of general Lie algebras

This section we devote to an introduction of Lie algebras, probably the most important algebras which one can ﬁnd in a large spectrum of possible applications. 2.5. THEORY OF GENERAL LIE ALGEBRAS 41

2.5.1 Lie algebras

Lie algebras get their name after Sophus Lie. They are algebras which have the most practical and theoretical use in studying geometrical objects such as Lie groups and differential manifolds. In this section we summarize the main notions and properties concerning Lie algebras. We begin with the general definition. A Lie algebra is a vector space A over a field K equipped with a bilinear Lie bracket product [ , ] which is anti-commutative and satisfies the Jacobi identity · · [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y]] = 0, (2.133) for all X, Y, Z A. ∈ For a given associative algebra A with multiplication , a Lie algebra L(A) can ∗ be constructed by setting a commutator in A equal to:

[X, Y] = X Y Y X. (2.134) ∗ − ∗ The Jacobi identity for commutators in L(A) is a consequence of the associativity of the multiplication . The associative algebra A is called an enveloping algebra ∗ of the Lie algebra L(A). In this fashion one can define the general linear Lie algebra gl(n, K) as a Lie algebra of all endomorphisms of Kn (i.e., represented as n n matrices over K), which together with the ordinary matrix product composes × an associative algebra over K. We note that every subalgebra of gl(n, K) is called a linear Lie algebra over K. It can be shown that every Lie algebra can be considered as an embedding into one that arises from an associative algebra [5]. Two Lie algebras L, L0 over K are isomorphic if there exists a vector space isomorphism φ : L L satisfying −→ 0 φ([X, Y]) = [φ(X), φ(Y)], X, Y L. (2.135) ∀ ∈ Then φ is called an isomorphism of Lie algebras. The notion of (Lie) subalgebra of L is defined similarly: A subspace A of L is called a Lie subalgebra if [x, y] A ∈ whenever x, y A; in particular, K is a Lie algebra in its own right relative to ∈ the inherited operations. Note that any non-zero element x L defines a one ∈ dimensional subalgebra Kx, with trivial multiplication, because any element of L commute with itself [40]. As examples of Lie algebras we can mention any commutative group, the cross product of 3-dimensional vectors, Lie algebras of Lie groups, Lie algebras generated by the commutator of vector fields, Lie derivatives and Kac-Moody algebras as an example of infinite-dimensional Lie algebras [5].

2.5.2 Matrix Lie algebras

As we already mentioned the space of n n matrices over K forms a Lie algebra, × because the matrix product is associative. Hence, it becomes apparent that every 42 CHAPTER 2. MATHEMATICAL PRELIMINARIES subgroup of Mat(n, K) closed under the action of the matrix commutator deﬁnes a Lie subalgebra of gl(n, K). Here we are interested in the case when K = R or C. We summarize the most important matrix Lie algebras in the list below. We notice that these algebras are identiﬁed as tangent spaces at the identity of corresponding matrix Lie groups.

1. The general linear Lie algebra gl(n, R)(gl(n, C)) of all real (complex) square n n matrices is the endomorphisms group Mat(n, R) (Mat(n, C)) endowed × with the ordinary matrix commutator.

2. The special linear Lie algebras sl(n, R), sl(n, C) are the spaces of all n n × matrices with zero trace in Mat(n, R), Mat(n, C), respectively.

3. The Lie algebras so(n, R), so(n, C) are the algebras of skew-symmetric n n × orthogonal matrices A with

AT = A, (2.136) − where A so(n, R) or so(n, C). ∈ 4. The pseudo-orthogonal Lie algebra o(p, q), 0 p, q, is the algebra of all real ≤ matrices A satisfying

Ag + gAT = 0, (2.137)

where g = (gij) is a real metric of type (p, q).

5. The unitary Lie algebra u(n) is the algebra of all skew-Hermitian matrices U: T U† := U¯ = U. (2.138) − 6. The special unitary Lie algebra su(n) is the algebra consisting of all skew- symmetric Hermitian matrices with trace zero:

U† := U, trU = 0. (2.139) −

7. The pseudo-unitary Lie algebra u(p, q) is the algebra of complex n n × matrices U obeying:

Ag + gA¯T = 0,

where g = (gij) is a pseudo-Hermitian metric of signature (p, q).

8. The special unitary Lie algebra su(p, q) is the algebra of matrices in u(p, q) with zero trace [6]. 2.5. THEORY OF GENERAL LIE ALGEBRAS 43

2.5.3 Structure constants, Bianchi identities, Killing-Cartan metric Structure constants

Let L be a ﬁnite dimensional Lie algebra and take a basis e , e2, ..., en for (the { 1 } vector space) L. By bilinearity the [ , ]-operation in L is completely determined · · once the values [ei, ej] are known. Since [ei, ej] L, we can write them as linear ∈ combinations of the basis vectors ei . { } Xn k k [ei, ej] = cijek = cijek. (2.140) k=1 k The coeﬃcients cij in these relations are called the structure constants of L (relative to the given basis). The antisymmetry ck = ck and the validity of the Jacobi identity of the Lie ij − ji algebra L imply the Bianchi identities:

m l m l m l cil cjk + cjl cki + cklcij = 0. (2.141) Under a change of basis the structure constants transform as a tensor of type i k l r s 1 k (2, 1): if e0j = ajei, then cij0 = crsai aj(a− )l . We note that systems, which form the structure constants of some Lie algebra over a ﬁeld K, form an algebraic set S, deﬁned by the above linear and quadratic equations that correspond to the skew-symmetry and the Jacobi identity. The general linear group GL(n, K) acts on S by the formulae above. The set of all structure constants relative to all bases of a given Lie algebra forms an orbit (set of all transforms of one element) under this operation. Conversely, the systems of structure constants in an orbit corresponds to unique Lie algebra up to an isomorphism. Thus, there is a natural bijection between orbits (of systems of structure constants) and isomorphism classes of Lie algebras (of dimension n) [41].

Killing-Cartan metric

Consider a Lie algebra g over a ﬁeld K. Every element x of g deﬁnes the adjoint endomorphism adj(x) (also written as adjx) of g based on the Lie bracket via

adjx(y) = [x, y]. (2.142) Now, suppose g is of ﬁnite dimension. The trace of the composition of two such endomorphisms deﬁnes a symmetric bilinear form

B(x, y) = tr(adj(x)adj(y)), (2.143) with values in K, the Killing form on g. In terms of the coordinates in some basis the Cartan form is expressed as

i s i l k s l s i k B(x, y) = tr((adjx)k)(adjy)i ) = clkx csi y = Blsx y , so Bls = clkcsi (2.144) 44 CHAPTER 2. MATHEMATICAL PRELIMINARIES

The Killing form of g is invariant, in the sense that it satisﬁes

B([x, y], z) + B(y, [x, z]) = 0, x, y, z g. (2.145) ∀ ∈ It is in particular invariant under any automorphism of g (for the proof see [42], Lemma 3.1). The Killing form is deﬁned as bilinear symmetric form. If additionally the Killing form is non-degenerate, it can be used as a metric. In particular, it can be shown that (Cartans second criterion) the Killing form of a Lie algebra g is non-degenerate if and only if the Lie algebra g is semisimple 6 [43, 42]. Remark: The map x adj assigns to each x in g an operator in End(g) 7−→ x preserving the linearity and the bracket, in the sense that it holds:

([adj , adj ])(z) = adj (z), x, y, z g. (2.146) x y [x,y] ∀ ∈ The proof can be easily obtained using the Jacobi identity. That is why the adjoint endomorphism adj is a representation of a Lie algebra g on a vector space g. For n g = K the adjx are matrices; expressed in terms of a basis e , ..., en of g we can { 1 } write for a g ∈ Xn [a, ej] = (adja)kjek. (2.147) k=1

The set of the matrices adja generates a n-dimensional representation of g, the adjoint representation of g [41].

2.5.4 Roots and weights

It turns out that the studying of Lie algebras and corresponding semisimple Lie groups as well as their classification is profoundly connected with the structures called root systems. In this section we only briefly touch the notion root and give a first outlook of the topic. To introduce the definition of roots of a Lie algebra we need some further concepts. At first we give the notion of the Cartan subalgebra. Let g be a Lie algebra. Then a Cartan subalgebra h is the maximal subalgebra of g which is self-normalizing, that is, if [g, h] h for all h h, then g h as well. Any Cartan ∈ ∈ ∈ subalgebra h is nilpotent, i.e.,

h = 0, for k 0, (2.148) Dk ≥ where is deﬁned inductively as D

1h = [h, h], kh = [h, k 1h]. (2.149) D D D − 6 That is if g is a direct sum of simple Lie algebras gi, that means gi should be non-abelian Lie algebras whose only ideals are 0 and gi itself. { } 2.5. THEORY OF GENERAL LIE ALGEBRAS 45

A subalgebra h of a semisimple Lie algebra g is called Cartan algebra if h is abelian, i.e., [h, h] = 0 and h = X g :[X, H] = 0, H h . adj is for each { } { ∈ ∀ ∈ } H H h diagonalizable. It can be shown that every semisimple Lie algebra owns ∈ a Cartan algebra [44]. The dimension of h is called the rank of g. It can be proved that for any two Cartan subalgebras h1, h2 of a Lie algebra g there exists an automorphism φ : g g such that φ(h ) = h2. −→ 1 Now we can proceed to the notion of a root. Let g be a ﬁnite dimensional Lie algebra over the complex numbers and let h be its Cartan subalgebra. It can be shown that then there exists a ﬁnite subset of Φ consisting of α h 0 such that ∈ ∗\{ } g is a direct sum of h-invariant7 vector spaces M g = gα, g0 = h, (2.150) α Φ 0 ∈ ∪{ } satisfying the property that for each α Φ the space g contains a common ∈ α eigenvector of h with eigenvalue α : h C, and this is the only eigenvalue of h −→ on g . Moreover, if we set g = 0 for α < Φ 0 , then for any α, β h we have α α ∪ { } ∈ ∗ [g , g ] g + (the proof can be found in [42], Theorem 4.5). In other words g α β ⊂ α β α can be written as

g = x g :[h, x] = α(h)x for all h h . α { ∈ ∈ }

The elements of Φ are called the roots of g with respect to h, and the gα are called the root spaces. The root decomposition (2.150) behaves nicely with respect to the Killing form, in the sense that if α, β Φ 0 are such that α + β , 0, then B(g , g ) = 0 ∈ ∪ { } α β and for x, y h we have ∈ X B(x, y) = α(x)α(y) dim gα. (2.151) α Φ ∈ For the proof look in [42], Lemma 4.7. The root decomposition (2.150) has especially nice properties when g is a semisimple Lie algebra. So from here for the rest of the section we take g semisimple. Then the Killing form is non-degenerate. This has many consequences:

1. Φ spans h∗;

2. dim g = 1, α Φ; α ∀ ∈ 3. the restriction of B( , ) to h is non-degenerate; for each linear form α on h · · there exists a unique element hα, called root vector, such that B(h, hα) = α(h), h h; ∀ ∈ 7That is, if [h, g ] g . α ⊂ α 46 CHAPTER 2. MATHEMATICAL PRELIMINARIES

4. Φ = Φ and [gα, g α] = Chα, α(hα) , 0; − − L 5. g = h gα g α is an orthogonal direct sum; ⊕ α Φ ⊕ − ∈ 6. for each α Φ the subspace hα = [gα, g α] h obeys hα ker(α) = 0 h; ∈ − ⊂ ∩ ⊂ (see [42, 41, 12]). Moreover, the following statements ([42], Lemma 5.4) hold:

2hα If α, β Φ, then β(Hα) Z, and β β(Hα)α Φ, where Hα = , α Φ. • ∈ ∈ − ∈ B(hαhα) ∈

If α + β , 0, then [g , g ] = g + . • α β α β For all α Φ and t Z we have tα Φ if and only if t = 1. • ∈ ∈ ∈ ± We now summarize some of the previous constructions. Let g be a semisimple Lie algebra over C with Cartan subalgebra h. Let α h be the set of roots of g, ∈ ∗ that is, the eigenvalues of the adjoint action of h on g. The non-degeneracy of the restriction of the Killing form of g to h allows us to deﬁne h h such that α ∈ α(h) = B(h , h), for any h h. It can be shown ([42], Lemma 5.4) that for any α ∈ α, β Φ the Cartan numbers, deﬁned as ∈

B(hα, hβ) nβ,α = 2 , (2.152) B(hαhα) are integers. Moreover, one can prove that for each α Φ a vector X g can ∈ α ∈ α be chosen such that for all α, β Φ ∈

[Xα, X α] = hα, [h, Xα] = α(h)Xα for h h; • − ∈ [X , X ] = 0, if α + β , 0 and α + β < Φ; • α β

[X , X ] = N X + if α + β Φ, • α β α,β α β ∈ 2 where the constants Nα,β satisfy Nα,β = N α, β. For any such choice N = − − − α,β q(1 p) − α(h ), where β + nα, p n q, n Z, is the series of roots containing β ([12], 2 α ≤ ≤ ∈ Theorem 5.5). We define hR as the vector space spanned by H defined above. Since n Z α α,β ∈ any root defines a linear map hR R and Φ h . The restriction of the Killing −→ ⊂ R∗ form of g to hR defines a metric on hR ([42], Lemma 6.1). As a consequence, one can get h = hR + ihR. Let us write ( , ) for the positive definite symmetric bilinear · · form on hR which corresponds to the Killing form:

(α, β) = B(hα, hβ) = α(hβ) = β(hα). (2.153)

Hence, we obtain (α, β) 2 = β(H ) = n Z. (2.154) (α, α) α α,β ∈ 2.5. THEORY OF GENERAL LIE ALGEBRAS 47

The pair (hR∗ , Φ) is an example of an abstract pair (V, R) consisting of a finite set R, called roots system, of non-zero vectors spanning a real vector spaceV equipped with a positive-definite symmetric bilinear form ( , ), such that any two distinct · · proportional vectors of R are negatives of each other, the reflections with respect (α,β) to the elements of R preserve R and 2 Z, for any α, β Φ. The dimension (α,α) ∈ ∈ of V is called the rank of R. The reflections in the roots of R generate a group of transformations called Weyl group, W = W(R)[42]. The importance of the concept of a root system is based on the fact that the isomorphism classes of complex semisimple Lie algebras bijectively correspond to equivalence classes of root systems8. A basis S R of a root system R is a ⊂ subset such that it is also a basis of the vector field V of R and every root of R is an integral linear combination of the elements of S all of whose coefficients have the same sign. The elements of S are called simple roots, and the elements of R that can be written as linear combinations of simple roots with positive coefficients are called positive roots (denoted as R+). The matrix of size r r, where r = S × | | is the rank of R, whose entries are the Cartan numbers nα,β, is called the Cartan matrix of R. Let ρ : g gl(V) be a representation. λ h is called a weight of ρ relative −→ ∈ ∗ to h if

Vλ := v V : ρ(H)(v) = λ(H)v H h (2.155) { ∈ ∀ ∈ } does not consist only of the zero vector. Non-zero vectors of Vλ are called weight vectors, Vλ is called weight space [5]. All weights of a representation ρ compose a set Γ(ρ) with Γ(ρ) , 0, Γ(ρ) < . If V is the adjoint representation of g, by | | ∞ the deﬁnition its weights except zero are roots of g, the weight spaces are root spaces, and weight vectors are root vectors. We call λ Γ the highest weight if ∈ λ + α < Γ for all α R+. Each v Vλ 0 then is called a maximal vector. It can be ∈ ∈ \{ } shown that for each representation ρ : g gl(V) there exists a highest weight. −→ Furthermore, we can write [45] M V = Vλ; (2.156) λ Γ ∈ λ λ+α + ρ(X )(V ) V α R , λ h∗. (2.157) α ⊂ ∀ ∈ ∈

The root system R V is irreducible if V cannot be written as an orthogonal ⊂ direct sum V = V V2 such that R = R R2, where Ri Vi, i = 1, 2, is a root 1 ⊕ 1 ⊂ ∈ system. It can be shown that any root system is uniquely determined by its Dynkin diagram (complete classiﬁcation see in [12]). Any irreducible root system is one

8 We call two root systems (V1, R1) and (V2, R2) equivalent if there exists an isomorphism φ : V V such that φ(R ) = R and for some constant c R∗ we have (φ(x), φ(y)) = c(x, y) for 1 −→ 2 1 2 ∈ any x, y V . ∈ 1 48 CHAPTER 2. MATHEMATICAL PRELIMINARIES

of the classical root systems An = sl(n), n 1, Bn = o(2n + 1), n 2, Cn = sp(2n), ≥ ≥ n 2, Dn = o(2n), n 3, or one of the exceptional root systems G2, F , E6, E7, E8 ≥ ≥ 4 ([42], Theorem 6.14). Complete consideration of the classical root systems can be found in [43]. For classical root systems we can assign a basis in terms of root vectors as follows:

An : e e2, e2 e3, ..., en en ; • 1 − − − +1

Bn : e1 e2, e2 e3, ..., en 1 en, en; • − − − −

Cn : e1 e2, e2 e3, ..., en 1 en, 2en; • − − − −

Dn : e1 e2, e2 e3, ..., en 1 en, en 1 + en. • − − − − −

2.5.5 Some elements of representation theory

Recall that a linear transformation φ : L L (L, L Lie algebras over a ﬁeld K) −→ 0 0 is called a homomorphism if φ([x, y]) = [φ(x), φ(y)], for all x, y L. A represen- ∈ tation of a Lie algebra L is a homomorphism φ : L gl(n, V)(V is vector space −→ over K), the number n is called the dimension of the representation. Although we require L to be ﬁnite dimensional, it is useful to allow V to be of arbitrary dimension, gl(n, V) makes sense in any case. The vector space V, together with the representation φ, is called an L-module. A representation is called faithful if its kernel is 0. A representation L gl(n, V) is called irreducible if the only L- −→ invariant subspace W V, W , V, is W = 0. Recall that a subspace W V, W , V ⊂ ⊂ is L-invariant if LW W [42, 40]. Any subalgebra is equipped with a natural rep- ⊂ resentation of dimension n = dim gl(L), namely with the adjoint representation already mentioned previous sections.

2.5.6 Deﬁnitions of afﬁne and Kac-Moody algebras

Let us briefly discuss the main concepts of the theory of infinite-dimensional Lie algebras, so-called Kac-Moody algebras. We begin with the theory of affine Lie algebras. They represent a class of infinite-dimensional Lie algebras which can be canonically constructed out of a finite-dimensional semisimple Lie algebra. Each affine Lie algebras is a Kac-Moody algebra whose generalized Cartan matrix is positive semi-definite and has corank 1. As we already mentioned every (untwisted) affine Lie algebra may be constructed from a finite-dimensional semisimple Lie algebra. By the definition semisimple Lie algebras are direct sums of simple Lie algebras. Therefore, affine Lie algebras built of semisimple Lie algebra are also direct sums of the affine Lie algebras constructed on these simple Lie algebras. In oder to simplify our discussion it will suffice to consider the construction for simple Lie algebras [5]. 2.5. THEORY OF GENERAL LIE ALGEBRAS 49

The way we construct the affine Lie algebra associated to a finite dimensional simple Lie algebra is the following. Let g be a finite dimensional simple Lie algebra over complex numbers C. We repeat that it is a non-Abelian finite- dimensional Lie algebra with no non-trivial ideals. We denote the Lie bracket 1 on g by [ , ]0. Let C[t, t ] denote the ring of Laurent polynomials in t C and · · − ∈ t 1 C, i.e., it consists of elements of the form − ∈ X ∞ n a = ant , an C. (2.158) n= ∈ −∞ We put 1 g˜t = g C[t, t− ]. (2.159) ⊗ n If f , ..., fa g denotes a basis for g, then we denote the element fs t by fs(n), { 1 } ∈ ⊗ where s = 1, ..., a and tn C[t, t 1], n Z, is in the canonical basis for C[t, t 1]. ∈ − ∈ − Then the set of vectors given by

n fs(n) = fs t : s = 1, ..., a; n Z (2.160) { ⊗ ∈ } generates a basis of g˜t. 1 We are interesting in the algebra g˜ appearing as a restriction of g˜t to t S = ∈ z C : z = 1 . In this case we can write t = eiθ, where θ [0, 2π], and g˜ then is { ∈ | | } ∈inθ spanned by fs(n) = fs en s = 1, ..., a; n Z , where en = e . The commutator { ⊗ | ∈ } of the Lie algebra g˜ is declared in natural way as

m+n [es(m), ep(n)] = [es, ep]0 t . (2.161) ∼ ⊗ The algebra g˜ has the name loop algebra. The reason for that is that g˜ can be identified with the space of smooth mappings from S1 to the Lie algebra g, which P inθ we denote by Lg and call loops. Moreover, we may identify G˜ g˜ with xne , ∈ n Z ∈ where xn in g tends to zero as n approaches infinity. Hence, we may write G˜ in | | g˜ as a Fourier series with coefficients xn in g [46]. Define g˜ = g˜t C c and g˜ = g˜ C c as the central extensions of g˜t and g˜, t0 ⊕ · 0 ⊕ · respectively. With respect to the bracket given by

[x(m), y(n)] = [x(m), y(n)] + < x, y > mδm+n,0c, [x, c] = 0, (2.162) ∼ where x(m), y(n) g˜t and g˜ respectively and < , > denotes the usual Cartan- ∈ · · Killing form on g, the algebras g˜t and g˜ are Lie algebras. Expressed in original notation, the commutator can be given as

m+n [x(m), y(n)] = [x(m), y(n)]0 t + < x, y > mδm+n 0c, (2.163) ⊗ , where x, y are basis vectors of g and n, m Z. The element commuting with all ∈ g g g elements of ˜t0 is called central and the Lie algebras ˜t0 and ˜0 are called affine Kac- g g Moody algebras associated with [5, 46]. We note that the algebra ˜t0 is generated 50 CHAPTER 2. MATHEMATICAL PRELIMINARIES by the elements x( 1), x(0), x(1) : x g and that the mapping x x(0) is a Lie { − ∈ } −→ g g algebra isomorphism from into ˜t0, i.e., it is invertible, linear and conserves the g g g bracket operation. Since ˜0 is the restriction of ˜t0 this naturally also holds for ˜0. Now for generalization of the notion of affine Lie algebras we turn to Kac- Moody algebras from which the affine algebras can be derived. Let A = (aij) be 9 a generalized Cartan matrix , i.e., a square n n-matrix such that aii = 2, aij 0 × ≤ for i , j, and aij = 0 implies aji = 0 . The associated Kac-Moody algebra g(A) is a complex Lie algebra constructed on 3n generators ei, fi, hi (i = 1, ..., n) and the following relations (i, j = 1, ..., n):

[hi, hj] = 0, [ei, fi] = hi, [ei, fj] = 0 if i , j, (2.164)

[hi, ej] = aijej, [hi, fj] = aij fj, (2.165) − 1 aij 1 aij (adjei) − ej = 0, (adj fi) − fj = 0 if i , j (2.166) where adj is again the adjoint representation of g [47, 48]. Another, but equivalent way to define a Kac-Moody algebra, is by using the so-called realization of a generalized matrix A and the notion of roots and dual roots, known as coroots. It turns out that both definition coincide, hence we only note that in the terms of the second definition the generator hi can be considered as basis vectors of the Cartan subalgebra h. The dual space h∗ then is spanned by h the vectors hi∗ defined via hi∗(hj) = aij. They compose a root system of . In this sense applying concepts of the foregoing sections the Kac-Moody algebra g can be diagonalized into weight eigenvectors which we define as earlier:

x h :[g, x] = ω(x)g, (2.167) ∀ ∈ where ω is an element of h . Then g g is said to have weight ω. The relations ∗ ∈ h (2.164) yield that the Cartan subalgebra has weight zero, ei has weight ai∗ and fi has weight a . If the Lie bracket of two weight eigenvectors is non-zero, then − i∗ its weight is the sum of their weights [5]. The classification of different types of Kac-Moody algebras breaks up into three subclasses. It is convenient to describe them assuming that the matrix A is symmetrizable, i.e., A can either be decomposed into DS, where D is diagonal with positive integer entries and S is a symmetric matrix, or not, i.e., there is no partition of the set 1, ..., n into two non-empty subsets so that aij = 0 whenever j belongs to the first subset, while j belongs to the second.Then there are the following three possibilities [5, 47]:

There is a vector v of positive integers such that all the coordinates of the • vector Av are positive. In such case all the principal minors of the matrix A are positive and the Lie algebra g(A) is ﬁnite-dimensional.

9 Note that ﬁnite-dimensional semisimple Lie algebras are Kac-Moody algebras since the Cartan matrix nαβ of a semisimple Lie algebra is obviously a generalized Cartan matrix. 2.6. ELEMENTS OF LIE GROUP THEORY 51

There is a vector v of positive integers such that Av = 0. In such a case all • the principal minors of the matrix A are non-negative and det A = 0; the algebra g(A) is infinite-dimensional. The Lie algebras of this subclass are called affine Lie algebras introduced in the first part of this section.

There is a vector v of positive integers such that all the coordinates of the • vector Av are negative. In such a case the Lie algebra g(A) is of indeﬁnite type.

2.6 Elements of Lie group theory

Here we consider the Lie theory from geometrical point of view and show explicitly the relation of Lie algebras with geometrical objects such as manifolds.

2.6.1 Matrix Lie groups and their relation with Lie algebras

We begin with a very important point concerning the theory of Lie algebras. Next we consider matrix groups which are tightly connected with the deﬁnition of any Lie algebra. It turns out that the structure of Lie groups is more complicated then the structure of the corresponding Lie algebra. However, it is possible to know the specialties of the underlying Lie group from the knowledge of the properties of the Lie algebra. All the Lie groups studying in this section are all subgroups (of a certain sort) of the general linear groups. Let Mat(n, C) denote the space of all n n matrices × with complex entries and let Gl(n, C) denote invertible matrices in Mat(n, C).

Deﬁnition 2. Let Am be a sequence of complex matrices in Mat(n, C). We say that Am converges to a matrix A if each entry of Am converges (as m ) to the corresponding −→ ∞ entry of A (i.e., if (Am)kl converges to Akl for all 1 < k, l < n).

Deﬁnition 3. A matrix Lie group G is any subgroup of GL(n, C) with the following property: If Am is any sequence of matrices in G and Am converges to some matrix A, then either A G, or A is not invertible. In other words, a matrix Lie group is a closed ∈ subgroup of GL(n, C) (most of the interesting matrix Lie groups G have the stronger condition - G is closed in Mat(n, C)).

Let us begin with the most important examples of matrix Lie groups. It turns out that we have already take a contact with them in another context as with transformation groups preserving a certain property or bilinear form.

The general linear group GL(n, K) consists of all invertible n n matrices • × over K (we will assume that K = R or C).

The special linear group SL(n, K) consists of the elements of GL(n, K) with • determinant 1. 52 CHAPTER 2. MATHEMATICAL PRELIMINARIES

The (real) orthogonal group O(n, R) or just O(n) consists of the real n n • × matrices M with MTM = Id; for the complex orthogonal group O(n, C) we replace real by complex in the deﬁnition.

The special (real) orthogonal group SO(n, R) =SO(n) is O(n) SL(n, R); sim- • ∩ ilarly for SO(n, C).

The unitary group U(n) consists of all the (complex) matrices M with • M M =Id; the special unitary group SU(n) is U(n) SL(n, C). † ∩ The symplectic group Sp(n, K) consists of all 2n 2n matrices over K with • × MT JM = J (where J is a matrix with J2 = Id); such matrices automatically − have det M = 1. The symplectic group Sp(n) is Sp(n, C) U(2n). ∩ Finally the Lorentz group consists of all real 4 4 matrices M with MTId M = • × 3,1 Id3,1 (Id3,1 is the identity matrix of signature (3, 1)) as an example of indefinite real orthogonal Lie group O(3, 1).

The set of all n n matrices over K can be identified with the standard vector × space of dimension n2 over K. Therefore, the groups defined above are subsets of various spaces Rm or Cm, defined by a finite number of simple equations. Mathematically, they are algebraic varieties (except for U(n) and SU(n)[41]) and consequently they are all topological differentiable manifolds. We now come to the relation of these groups with the corresponding Lie algebras. It is customary to use lowercase Gothic characters such as g for a Lie algebra to refer to the corresponding Lie group G. Briefly we can formulate the correspondence between Lie algebras and Lie groups by stating that a Lie algebra is the tangent space of a Lie group at the unit element [41,6]. In other words, for a given a Lie group a Lie algebra can be associated to it by endowing the tangent space to the identity with the differential of the adjoint map. So the elements of the corresponding Lie algebra are tangent vectors to the curves through the identity of G. There are two deep theorems which express the connection of Lie algebras and Lie groups and their general structure. The first one is called Ado’s theorem. One version of it says that every finite-dimensional Lie algebra is isomorphic to a matrix Lie algebra [49, 45]. For every finite-dimensional matrix Lie algebra, there is a linear group (matrix Lie group) with this algebra as its Lie algebra. So every abstract Lie algebra is the Lie algebra of some (linear) Lie group. This is in contrast to the situation for Lie groups, where most, but not all, Lie groups are matrix Lie groups. The second important theorem is Lie’s fundamental theorem. It describes a relation between Lie groups and Lie algebras. In particular, any Lie group gives rise to a canonically determined Lie algebra, and conversely, for any Lie algebra there is a corresponding connected Lie group (Lie’s third theorem, 2.6. ELEMENTS OF LIE GROUP THEORY 53 the prove is given in [45]).This Lie group is not determined uniquely, however, any two connected Lie groups with the same Lie algebra are locally isomorphic. The correspondence between Lie algebras and Lie groups can be used in the classification of Lie groups and in the related question of the representation theory of Lie groups. Every representation of a Lie algebra determines uniquely a representation of the corresponding connected simply connected Lie group, and conversely every representation of any Lie group induces a representation of its Lie algebra; the representations are in one-to-one correspondence [5].

2.6.2 Generic deﬁnition of groups

We used in previous sections the notion of a group. We have defined the orthogonal, linear etc. groups and meant always a set whose elements are united in a certain way. It is time to give a more general definition of this object. A group G is a set of elements together with a map : G G G that · × −→ combines any two elements a and b of G into another element denoted by a b. The · operation “ ” is an arbitrary map specified for any given group. For example one · can take addition for the group of integer. The only condition on this operation is that the pair (G, ) must satisfy four requirements known as the group axioms: · Closure: if a, b G then a b G; • ∈ · ∈ Associativity: a (b c) = (a b) c holds for all a, b, c G; • · · · · ∈ Existence of identity: there exists an element e G such that a e = e a = a, • ∈ · · a G; ∀ ∈ Existence of inverse: for any a G there is in G an element a 1, which obeys • ∈ − a a 1 = a 1 a = e. · − − · Examples are the real numbers with the ordinary addition (R, +), symmetry groups, cyclic groups etc. (see [5]). For a given group G together with a binary operation “ ”, we say that some · subset H of G is a subgroup of G if H also forms a group under the operation “ ”. ·

2.6.3 Topological groups and smooth Lie groups

We have mentioned that any Lie algebra is a tangent space of a Lie group at a given point. The concept of tangents spaces is related with topological objects called manifolds. To go stepwise to the consideration of Lie groups as manifolds we ﬁrst of all endow the group with certain topological properties.

Deﬁnition 4. A topological group is a group G together with a topology on G (see 2.1.9) that satisﬁes the following two properties: 54 CHAPTER 2. MATHEMATICAL PRELIMINARIES

The map p : G G G deﬁned by p(g, h) = gh is continuous when G G is • × −→ × endowed with the product topology.10

The map inv : G G defined by inv(g) = g 1 is continuous. • −→ − The first property is equivalent to the statement that, whenever U G is ⊆ open, and g g2 U, there exist open sets V , V2 such that g V , g2 V2 and 1 ∈ 1 1 ∈ 1 ∈ V V2 = h h2 : h V , h2 V2 U. The second property is equivalent to that, 1 { 1 1 ∈ 1 ∈ } ⊆ whenever U G is open, the set U 1 = g 1 : g U is open. Moreover, we ⊆ − { − ∈ } remark that every subgroup of a topological group, endowed with the subspace topology, is a topological group [50, 51]. At this place we give some examples of topological groups: (R, +), (R 0 , ), GL(n, R) and GL(n, C)[50]. \{ } · Now we turn to the idea of the Lie groups. A Lie group is a group whose group structure and differentiable structures are compatible, i.e., the multiplication map and the inversion map are differentiable. In this sense a Lie group is a differentiable manifold11, on which the group operations of multiplication and inversion are differentiable maps. Differentiability of the group multiplication means that the map µ : G G G, µ(x, y) = xy with x, y G is a differentiable × −→ ∈ mapping from the product manifold G G into G. It is convenient to assume × that Lie group is an infinite times continuously differentiable manifold. We then say G is smooth. All of the previous examples of groups fall within the class of smooth Lie groups. We do not list them here. The representation of a Lie group G on finite dimensional vector space V (real or complex) is a homomorphism ρ : G GL(V) of Lie groups. The adjoint −→ representation of a Lie group G is the natural representation of G on its own Lie algebra given by the map adj : G Aut(g), g (adj (e)) sending each 0 −→ 7−→ g 0 element g of the group to the derivative of the adjoint map at the identity.

2.6.4 Left, right and adjoint action of a Lie group G on itself

Let G be a group. Define La : G G by La(g) = ag and Ra : G G by −→ −→ Ra(g) = ga. We call La the left action of G on itself and Ra the right action of G on itself given by the element a. The adjoint action on G is defined as combination 1 of both these actions, precisely as adjg : G G, adg(h) = ghg . −→ − We call a vector field X on G left invariant if it is invariant under any left action La of G (respectively right action). This can be written with the help of pushforward transformation: (La) Xa = Xaa for all a, a0 G. Here (La) is ∗ 0 0 ∈ ∗ equal to the differential of La. A left (right) invariant vector field is always

10Open sets on G G are explained intuitively: that are subsets of G G which can be written × × as O O0, where O, O0 G are open sets relative to the topology on G. × ∈ 11A differentiable manifold is a topological, paracompact, Hausdorff space which is locally homeomorphic to Euclidean space and whose transition maps are all differentiable (more information can be found for example in [5,6, 14]. 2.7. HOMOGENEOUS SPACES 55 differentiable. The Lie algebra of G can then be identified with the set of all left invariant vector fields together with usual addition, scalar multiplication and usual bracket operation. As a vector space the algebra g then is isomorphic to the tangent space at the identity Te.

2.7 Homogeneous spaces

In this section we concentrate on a special type of Lie groups, namely on the homogeneous spaces. Furthermore, we investigate and get closer with the notion of nearly Kahler¨ coset spaces which are of a particular interest of this thesis.

2.7.1 Closed subgroup H of a Lie group G

We consider a subset H of a topological group G which is closed as a subset, i.e., the complement of H is open relative to the topology on G. Let now G be a Lie group. A submanifold H of G is called a Lie subgroup if the following conditions hold:

1. H is a subgroup of the group G;

2. H is a topological group.

It can be shown that a Lie subgroup itself is a Lie group. Moreover, if H is a Lie subgroup of a Lie group G, then the Lie algebra h of H (i.e., it is the tangent space of H at the identity) is a subalgebra of g which is the Lie algebra of G (see [12], Theorem 2.1). Remark: It can be immediately seen that H is a closed subgroup of G and it is a Lie group. The converse to this result is also true, but harder to prove: every closed subgroup H of a Lie group G is a submanifold of G [52, 45].

2.7.2 Homogeneous spaces G/H and H G of a Lie group as equiv- \ alence classes

An action of a Lie group G on itself defines of course an action of any subgroup H G. In particular, the left and right actions define an action of any subgroup ⊂ H G on the whole group G. The orbits of the transformation group obtained in ⊂ this way are called respectively the left and the right cosets of G under H. Thus, a left (right) coset consists of all elements of the form hg (gh), where g G is ∈ some fixed element, and h takes all possible values in H. We denote the cosets respectively by

Hg = hg : h H and gH = gh : h H . (2.168) { ∈ } { ∈ } 56 CHAPTER 2. MATHEMATICAL PRELIMINARIES

The general property of the left (right) cosets spaces follows from the fact that each element of G is in only one left (right) coset. Hence, the distribution of the elements among the left (right) coset spaces is disjoint. Furthermore, cosets can be considered as equivalence classes. That means that any two elements x, y of a left coset of H in G are related in a way that also x 1 y H (that holds if and only if the elements x, y are in the same left coset). We − ∈ then say that the two elements x, y of G are equivalent 12 and write x y. It is ∼ straightforward to see that this equivalence relation is an equivalent definition of left cosets. In this sense we say that the left cosets gH of H in G form equivalence classes relative to the above equivalence relation. We denote the set of all left (right) cosets by G/H (H G). \ We call a subgroup N of G normal if for all n N and all g G also gng 1 N, ∈ ∈ − ∈ i.e., in particular, if gN = Ng for all g G. In this case the set of all cosets form a ∈ group which we call the quotient group G/N [15,5]. For a Lie subgroup N which is not normal, the space G/N of left cosets does not have the structure of a group, but simply is a differentiable manifold on which G acts. Suppose G is a Lie group and H is a closed subgroup of G. Let π : G G/H −→ denote the natural mapping g gH of G onto G/H. The mapping π endows 7−→ the set G/H with the natural topology coming from G, since π is continuous and open.13 Hence G/H is a smooth manifold called coset manifold. Furthermore, the map π has the property that any two points in G/H can be joined by the action of G, i.e., the action is transitive (for all A, A G/H there exists a g G 0 ∈ ∈ such that gA = A0). We call the resulting manifold furnished with this transitive action a homogeneous space (see [12, 53]). In general a homogeneous space is a manifold M with a transitive action of a Lie group G. An equivalent definition is that a homogeneous space is a manifold of the form G/H, where G is a Lie group and H is a closed subgroup of G. It can be shown that the dimension of G/H is dim G dim H [53]. Examples are − the sphere Sn O(n + 1)/O(n) and the complex projective space CPn U(n + ' ' 1)/U(n) U(1). Independently of the fact that the definition of homogeneous × spaces was introduced by us for left coset spaces, it is quite obviously that the analogous consideration can be done also for the right cosets.

2.7.3 Splitting of the Lie algebra Lie G = Lie H (Lie H)⊥ ⊕ Let G be a Lie group, H its closed subgroup. Denote by g and h the Lie algebras corresponding to G and H, respectively. Due to the Cartan theorem, every closed subgroup H of a Lie group G is a Lie subgroup. That means in particular that H is a smooth manifold embedded in G. Since the Lie algebra g assigned to G

12In general an equivalence “ ” relation is a binary operation between two elements x, y of a ∼ set G which is reﬂexive x y, symmetric x y y x and transitive x y, y z x z. ∼ ∼ ⇒ ∼ ∼ ∼ ⇒ ∼ 13This means that images of open sets in G are open in G/H. 2.7. HOMOGENEOUS SPACES 57

is the tangent space TeG at the identity e of G and since that identity e is also the identity of the subgroup H, one can immediately see that the corresponding algebra h of H is a subspace of the tangential space of G: h = TeH TeG = g. ⊂ Consequently the Lie algebra h is a Lie subalgebra of g. Considering h and g as vector spaces this relation can be expressed as a direct sum:

g = h m, (2.169) ⊕ where m is a complement of h in g. The homogeneous space G/H is called reductive if the Lie algebra g can be decomposed into a vector space direct sum of the h and an adj(H)-invariant subspace m, that is, if

g = h + m; h m = 0; • ∩ adj(H)m m (i.e., [h, m] m). • ⊂ ⊂

The space m can be identiﬁed with the tangent space Te(G/H). It can be shown that if G is compact or connected and semisimple then G/H is reductive [54, 55, 56]. In particular, if G is semisimple the Lie algebra g can be decomposed g = m h with m = h , where h is the orthogonal complement of h relative to ⊕ ⊥ ⊥ the Killing form of g [57].

2.7.4 Splitting of commutators

Suppose, we have a ﬁxed decomposition g = h m of the Lie algebra g of a ⊕ compact semisimple Lie group G in its Lie subalgebra h of a Lie subgroup H G ⊂ and the complement m. Then the original commutator of g can be splitted in two subsets in the following manner. The generators eA , A = 1, ..., dim g, of { } the algebra g split in ei , i = 1, ..., dim h, and ea , a = 1, ..., dim m, in such way { } { } that the generators ei span the subalgebra h and the generators ei span the { } { } complement m. In this fashion the commutation relations

C [eA, eB] = fABeC (2.170) can also be rewritten in the explicit form

k k c i c [ei, ej] = fijek, [ei, ea] = fiaek + fiaec, [ea, eb] = fabei + fabec. (2.171)

In a special case when the coset space G/H is a reductive homogeneous space k the structure constants fia vanish and the above relations simplify to

k c i c [ei, ej] = fijek, [ei, ea] = fiaec, [ea, eb] = fabei + fabec. (2.172) 58 CHAPTER 2. MATHEMATICAL PRELIMINARIES

2.7.5 The Maurer-Cartan equations

If G is a Lie group and γ = γ(t) is a curve with γ(0) = A G, then the curve A 1γ ∈ − passes through the identity at t = 0. Hence, we conclude that the derivative 1 A− γ0(0) is a tangent vector at the identity, i.e., an element of the Lie algebra g, assigned to G. In this way we obtain a linear diﬀerential form, θ : G g, which −→ assigns to every element in G an element in g. In case if G is a subgroup of the general linear group Gl(n), the form θ can be written as

1 θ = A− dA. (2.173)

For simplicity we will work with this case, although the main theorem, the Maurer-Cartan equation, holds for any arbitrary Lie group. We note that the deﬁnition of the Lie groups amounts to constraints on A. The equation (2.173) gives a description of the Lie algebra g. For example, G =O(n) T 1 1 T yields AA =Id. Diﬀerentiating we obtain A− dA + (A− dA) = 0. Hence, g consists of antisymmetric matrices. Returning to our consideration, we take the derivative of the equation (2.173). 1 1 For that purpose we have to compute d(A− ): using d(AA− ) = 0, we obtain

1 1 dAA− + Ad(A− ) = 0 (2.174) d(A 1) = A 1dAA 1. (2.175) ⇒ − − − − Putting this in the equation (2.173), we get

1 1 1 1 1 dθ = d(A− ) dA = ( A− dAA− ) dA = A− dA A− dA, (2.176) ∧ − ∧ − ∧ which yields the formula known as the Maurer-Cartan equation

dθ = θ θ. (2.177) − ∧ This relation can also be formulated with the help of a left invariant 1-form ω on G. First of all, a diﬀerential form is called left invariant if (La)∗ω = ω, for all a G, or equivalently (La)∗ωa = ω 1 , where (La)∗ω is the pullback of ω deﬁned ∈ 0 a0− a via (L ω)x(X) = ω)ax((dLa)xX), x G, X TxG. Moreover, the set of all left a∗ ∀ ∈ ∈ invariant 1-forms constitutes the dual vector space of the algebra g. Then the Maurer-Cartan equation can be written as dω(X, Y) = 1 ω([X, Y]) for all ω g − 2 ∈ ∗ and X, Y g [5, 58, 43, 59]. ∈

2.7.6 Metric, Levi-Civita connection, afﬁne connection and torsion Metric

Since a Lie group G is a smooth manifold, we can endow G with a Riemannian metric. Among all Riemannian metrics on a Lie group, those for which the left 2.7. HOMOGENEOUS SPACES 59 translations (or the right translations) preserve the metric play an important role, because they respect the group structure G. Let g be a Riemannian metric on a Lie group G, i.e., a positive deﬁnite non- degenerate symmetric 2-form TG TG R. The metric g is called left-invariant × −→ (resp. right-invariant) iﬀ

g(x, y)a = g((dLa)bx, (dLa)b y)ab, (2.178)

(resp. g(x, y)a = g((dRa)bx, (dRa)b y)ba), (2.179) for all a, b G and all x, y T G. If a Riemannian metric is both left- and ∈ ∈ b right-invariant, then we call it biinvariant. It can be shown that left-invariant metrics on G correspond bijectively to scalar products on m (we again assume that g = h m). Since left-invariant translations are isometries and act transitively ⊕ on G, the space G equipped with a left-invariant metric is called a homogeneous Riemannian space [12, 58]. It was proven by Elie´ Cartan that for any compact Lie group G and any compact subgroup H G for the homogeneous space ⊂ G/H there exists a left-invariant Riemannian metric d, i.e., such that d(x, y) = d(gx, gy), x, y G/H, g G, where G acts naturally on G/H [60, 58]. In this case ∀ ∈ ∈ the group G itself admits a biinvariant metric.

Afﬁne connection

An aﬃne connection on a manifold M is an operation which assigns to each ∇ tangent vector X TM a linear map X of the vector space TM into itself, ∈ ∇ satisfying the following conditions:

= f X + g Y, X( f Y) = f X(Y) + (X f )Y. (2.180) ∇ f X+gY ∇ ∇ ∇ ∇

We call the expression XY the covariant derivative of Y in X direction. Let us ∇ look at how connection can be viewed locally. For this reason we rewrite the deﬁning equation (2.180) for basis vectors ea of TM: { } i b ea = dx ω e , (2.181) ∇ ⊗ ia b b i where ωαa(x) are some functions and x are the local coordinates on M. Consider the one-forms b i b ωa = dx ωia. (2.182) Then b ea = ω e . (2.183) ∇ a ⊗ b It can be shown that any transformation of the basis vectors assigned to every 1b point of M of the form ea0 = S− aeb leads to the transformation of the one-forms ω, in particular, b 1c d b 1c b ωa0 = S− aωc Sd + dS− aSc, (2.184) 60 CHAPTER 2. MATHEMATICAL PRELIMINARIES

Omitting the indices in the last equation and taking the exterior derivative of it give the transformation formula

1 dω0 ω0 ω0 = S− (dω ω ω)S. (2.185) − ∧ − ∧ Hence, we see that the form dω ω ω transforms as a tensor. We call the − ∧ corresponding 2-form Rb = dωb ωc ωb, i.e., a form with values in T M TM, a a − a ∧ c ∗ ⊗ the local curvature form of connection . ∇ We are interested in affine connections on Lie groups. That is why we define a left-invariant connections as next step. Let G be a Lie group together with , ∇ an affine connection on it. is said to be left-invariant if each left transition Lg ∇ (g G) is an affine transformation14 on G [12]. In the same way we can introduce ∈ a left-invariant connection on a reduced homogeneous space G/H if for all ∇ a G the functions τ(a): G/H G/H, xH axH are affine transformations ∈ −→ 7−→ [57]. The most important example of an affine connection is the Levi-Civita connection. It is the only connection which we say preserves the metric, i.e., it obeys X(g(Y, Z)) = g( LCY, Z) + g(Y, LCZ), (2.186) ∇X ∇X where g is a metric on G and X, Y, Z TG, and is torsion-free: ∈ LCY LCX [X, Y] = 0, (2.187) ∇X − ∇Y − where the left hand side of the equation (2.187) defines the torsion T Γ(T M ∈ ∗ ⊗ T M TM): T = Ti ∂ , where Ti = 1 Ti dxj dxk in some holonomic basis. In ∗ ⊗ ⊗ ∂xi 2 jk ∧ some non-holonomic basis ea the relation of Ta to the connection 1-form ω can { } be expressed by the following relation

dea + ωa eb = Ta. (2.188) b ∧ The torsion vanishes for the Levi-Civita connection. So that explicitly the Levi- Civita connection can be given by the formula:

1 g( LCY, Z) = X(g(Y, Z)) + Y(g(X, Z)) Z(g(X, Y)) ∇X 2{ − +g([X, Y], Z) + g([Z, X], Y) g([Y, W], X) . (2.189) − } In holonomic basis the components of the corresponding 1-forms ω, which we denote by Γ, can be expressed in the terms of the metric components:

1 Γk = gkl(g + g g ), (2.190) ij 2 il,j lj,i − ij,l

∂g g ij where ij,l = ∂xl [5]. 14 A diﬀeomorphism Φ of G is called an aﬃne transformation of G if is invariant under Φ, 1 ∇ i.e., if X(Y) = (Φ− )∗( X(Φ∗Y)), for all X, Y TG [12]. ∇ ∇Φ∗ ∈ 2.7. HOMOGENEOUS SPACES 61

It can be shown that on a reductive homogeneous spaces G/H with the decomposition g = h m there exists one and only one G-invariant connection ⊕ m which is torsion-free and such that the curves γ(t) = exp(tX) on G, for all ∇ X m, project by π : G G/H into geodesics15 γ (t) = π(γ(t)) in G/H. This ∈ −→ ∗ connection is called the canonical connection of the first kind [57] or the natural torsion-free connection. It coincides with the Levi-Civita connection on the coset m 1 space G/H. Using the above notation it can be given by Y = [X, Y]m for all ∇X 2 X, Y m Te(G/H)[54, 58]. ∈ Another important kind of affine connection on reductive homogeneous spaces is the Riemannian connection. It provides the same geodesics as the natural torsion-free connection, but has non-vanishing torsion. It can be expressed in terms of a adj(H) invariant non-degenerate symmetric bilinear form − B on m and the corresponding G-invariant metric g on G/H which is given by B(X, Y) = g(X, Y)0 for X, Y m. In this fashion the Riemannian connection can ∈ be written as 1 0 Y = [X, Y] + T(X, Y), (2.191) ∇X 2 where T(X, Y) is the symmetric bilinear map from m m into m, defined by × 2B(T(X, Y), Z) = B(X, [Z, Y]m) + B([Z, X]m, Y), for X, Y, Z m. This connection ∈ matches with the natural torsion-free connection if B(X, [Z, Y]m)+B([Z, X]m, Y) = 0 [58].

2.7.7 Kahler¨ coset spaces G/H

Let us start with a connected Lie group G. Let be H a closed subgroup of G and G/H be the space of left cosets of G modulo H, on which G acts by the left transitions. We assume that G is effective on H, i.e., H contains no non-trivial subgroup invariant in G. We define a complex smooth manifold as a manifold whose charts are homeomorphic to the open subsets of Cn and such that the transition functions are holomorphic. We can define a Kahler¨ manifold M as an n-dimensional complex manifold, i.e., a 2n-dimensional real manifold with a complex structure J, together with a Riemannian metric g on M, which is Hermitian16 in the sense that it satisfies:

g(JX, JY) = g(X, Y) X, Y TM (2.192) ∀ ∈ and whose Kahler¨ form ω(X, Y) = g(JX, Y) is closed: dω = 0 [14, 61].17

15 A geodesic on a smooth manifold M with an affine connection is defined as a curve ∇ γ(t) such that parallel transport along the curve preserves the tangent vector to the curve, so t γ˙(t) = 0 for all t. γ˙ denotes here the derivative of γ with respect to the curve parameter t. Is ∇γ˙( ) γv the unique geodesic, which for a given point p M and for a given tangential vector v TpM ∈ ∈ satisfies γv(0) = p and γ˙v(0) = v, then the exponential map is defined as expp(v) = γv(1). 16 A complex manifold always admits a Hermitian metric [14]. 17 C Setting Z = X + iY, W = U + iV TpM , one can extend the Riemannian metric g to ∈ 62 CHAPTER 2. MATHEMATICAL PRELIMINARIES

Now we can consider a complex manifold which is additionally a homogeneous space. We say that a coset space G/H is homogeneous complex (resp. homogeneous Kahler)¨ if it carries a complex analytic structure J (resp. Kahler¨ structure ω) invariant under the group structure G. For a compact connected Lie group it can be shown that “homogeneous complex and Kahlerian”¨ implies “homogeneous Kahlerian”¨ [61].

2.7.8 Nearly Kahler¨ coset spaces in d = 6

Let us consider a manifold with a weaker condition than existence of a complex structure on it. More concretely, we introduce an almost complex structure. An almost complex structure over a manifold is a smooth tensor field J of rank (1, 1) such that J2 = Id, where we regard J as a vector bundle isomorphism J : − TM TM. Then an almost Hermitian manifold (M2n, g, J) is a 2n-dimensional −→ real manifold, which admits an almost complex structure J compatible with a Riemannian metric g, i.e X, Y TM, g(JX, JY) = g(X, Y). In particular, every ∀ ∈ complex manifold is almost complex. A nearly Kahler¨ manifold is a special type of almost Hermitian manifolds. Namely, it is an almost Hermitian manifold such that the pair (g, J) satisfies the equation ( X J)X = 0, X TM, where denotes the Levi-Civita connection ∇ ∀ ∈ ∇ associated to metric g. It is called strict nearly Kahler¨ if ( X J) , 0 whenever ∇ X TM, X , 0. An important observation is that in this definition the last ∈ condition can be exchanged with one of following equivalent statements:

The canonical Hermitian connection uniquely defined by • ∇ 1 XY = XY + ( X J)JY, (2.193) ∇ ∇ 2 ∇ for all X, Y TM, has totally antisymmetric torsion. ∈ The Kahler¨ form ω related to g and J, defined via ω(X, Y) = g(JX, Y), obeys • 1 X TM, Xω = ιXdω, (2.194) ∀ ∈ ∇ 3 where ι denotes the inner product operating on the differential forms of a p p 1 manifold as ιX : Ω (M) Ω − (M), ιXω(X1, ..., Xp 1) = ω(X, X1, ..., Xp 1). −→ − − The form dω is of type (3,0)+(0,3) and the Nijenhuis tensor N, defined by • NJ(X, Y) = [X, Y] + J([JX, Y] + [X, JY]) [JX, JY], is totally skew-symmetric. − g(Z, W) = g(X, U) g(Y, V) + i(g(X, V) + g(U, Y)). − With respect to the basis ∂ and ∂ the metric components obey g = g and g = g . If g is { ∂zi } { ∂z¯i } µν¯ µν¯ µν µ¯ν¯ Hermitian, then it has the form g = g dzµ dzν¯ + g dzµ¯ dzν. µν¯ ⊗ µν¯ ⊗ 2.7. HOMOGENEOUS SPACES 63

We again are interested in coset spaces. So we say that a coset space G/H is homogeneous nearly Kahler¨ if the nearly Kahler¨ structure is invariant under the action of G. A classiﬁcation of homogeneous nearly Kahler¨ manifolds is not an easy exercise, nevertheless in the recent times there were reached some interesting results. At this point we refer the reader to [62,5]. We note that it was proven that for nearly Kahler¨ manifolds a splitting theorem holds which states that any nearly Kahler¨ manifold is locally a Riemannian product of a Kahler¨ manifold and a strictly nearly Kahler¨ one [63, 64]. Nearly Kahler¨ manifolds in 6 dimensions have a particular nature. First of all, for each almost Hermitian 6-manifold (M, J, ω), with ω( , ) = g(J , ), the · · · · following conditions are equivalent:

1. The form ω Λ1(M) Λ2(M) is non-zero and totally skew-symmetric ∇ ∈ ⊗ (that is, ω is a 3-form). ∇ 2. The structure group of M admits a reduction to SU(3), that is, there is a (3, 0)-form Ω with

Ω = 1, dω = 3λReΩ, dΩ = 2λω2, (2.195) | | − where λ is a non-zero real constant.

Furthermore, any nearly Kahler¨ manifold in 6 dimensions is isomorphic to one of the 4 homogeneous nearly Kahler¨ manifolds: S3 S3 =SU(2)3/SU(2), 6 3 × 3 G2/SU(3) S , Sp(2)/SU(2) U(1) CP , SU(3)/U(1) U(1) F . Their special ' × ' × ' property is also that they own a Killing spinor, i.e., on every nearly Kahler¨ 6- manifold M there exists a Killing spinor Ψ in the bundle of spinors G which satisﬁes

XΨ = λXΨ, X TM. (2.196) ∇ ∀ ∈ Here we used the Cliﬀord multiplication map TM G G. ⊗ −→ The consequence of the existence of a Killing spinor is that the manifolds listed above are Einstein manifolds, i.e., their Ricci tensor is proportional to the metric with Einstein constant λ 2 > 0. Furthermore, nearly Kahler¨ manifolds | | provide a natural example of almost Hermitian manifolds admitting a Hermitian connection with totally skew symmetric torsion. From this point of view they are of interest in string theory [65, 62].

Chapter 3

Yang-Mills-theory on nearly Kahler¨ manifolds

3.1 Yang-Mills equation on manifold M

We will now discuss the main aspects of connection on ﬁbre bundles and ﬁnally turn to the Yang-Mills theory which as we will see appears in a natural way if we restrict ourselves to a discussion of the topology of principal bundles. The major ideas as well as advanced treatments of this topic can be found in references [14, 58, 66, 67, 68, 69].

3.1.1 Principal ﬁbre bundle

Let M be a manifold and G an arbitrary Lie group. We define a (differentiable) principal fibre bundle over M with G as a pair of a manifold P and an action of G on P obeying the following conditions:

1. G acts freely on P on the right: (u, a) P G ua = Rau P; ∈ × −→ ∈ 2. M is the quotient space P/G of P given by the equivalence relation induced by G, and the canonical projection π : P M is differentiable; −→ 3. P is locally trivial, that is if every point x of M has a neighborhood U such that π 1(U) is isomorphic to U G in the sense that there is a diffeomorphism − × ψ : π 1(U) U G, such that ψ(u) = (π(U), φ(u)), where φ is a mapping − −→ × of π l(U) into G satisfying φ(ua) = (φ(u))a for all u π 1(U) and a G. − ∈ − ∈ The principal fibre bundle is usually denoted by P(M, G). P is called the total space or the bundle space, M the base space, G the structure group and π the projection. In particular, for each point x M, the set π 1(x) is a closed submanifold ∈ − 65 66 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

1 of P, called the fibre over x. One can see that every fibre π− (x) is diffeomorphic to G.

3.1.2 Transition functions

Consider now an open covering U of M. Then the concept of transition { α} functions makes it possible to consider a principal bundle P(M, G), which until now was a local object, as a global construction. Namely, because of the local 1 triviality each π− (Uα) is endowed with a diffeomorphism u (π(u), φα(u)) of 1 −→ π− (Uα) on Uα G such that φα(ua) = (φ(u))a. This means that if the image of × 1 1 u P under the projection π is in Uα Uβ, then φβ(ua)(φα(ua))− = φβ(a)(φα(a))− . ∈ ∩ 1 The last equation shows that the term φβ(a)(φα(a))− depends only on π(u) and not on u. Thus, the right multiplication is defined without reference to the local trivializations. Setting ψ : U U G as ψ (π(u)) = φ (u)(φ (u)) 1, we αβ α ∩ β −→ αβ β α − define a family of transition functions ψαβ of the bundle P(M, G) corresponding to the open covering U of M. It can be easily verified that { α} ψ = ψ ψ for x U U U . (3.1) γα γβ βα ∈ α ∩ β ∩ γ Finally, let U M be an open set of M. A smooth map s : U P(M, G) ⊂ −→ from the base space M into the bundle space P is called a local cross section if π(s(x)) = x, for all x U. If a section can be defined on the whole manifold M, ∈ then we call it a global cross section. The set of all cross sections of P is denoted by Γ(M, P).

3.1.3 Associated ﬁbre bundle

Given a principal fibre bundle P(M, G) and F a manifold, on which G acts on the left: G F (a, f ) a f F, we may construct an associated fibre bundle as × 3 −→ ∈ follows. Define an action of g G on P F by ∈ ×

1 (u, f ) (ug, g− f ), (3.2) −→ where u P and f F. Then the quotient space P G F := (P F)/G with respect ∈ ∈ × × to this group action is the associated fibre bundle E(M, F, G, P). Assume now that P(M, G) is a principal fibre bundle and H a closed subgroup of G. The action of G on the coset space G/H on the left is defined in a natural way. So let us consider the associated fibre bundle E(M, G/H, G, P) with standard fibre G/H. Since H is a subgroup of G, it acts on P on the right. Denoting the quotient space relative to this action P/H, one can show that it can be identified with the associated fibre bundle E = P G (G/H). × 3.1. YANG-MILLS EQUATION ON MANIFOLD M 67

3.1.4 Associated vector bundle

Let us specify the case when F is a k-dimensional vector space V. Let ρ be the k-dimensional representation of G on V. Then we deﬁne the associated vector bundle P V as an equivalence class (P V)/G, where we identify the points ×ρ × (u, v) and (ug, ρ(g) 1v) of P V, for each g G. − × ∈

3.1.5 Connection

Let us now devote ourself to the theory of connections on a principal bundle. For that purpose we use the splitting of tangent the space Tu(P) into vertical and horizontal subspaces. For a given principal bundle P(M, G) over a manifold M and a structure group G let Tu(P) be a tangent space of P at u P and Vu be the subspace of Tu(P) ∈ consisting of vectors tangent to the ﬁbre through u. A connection Γ on P is an assignment of a subspace Hu of Tu(P) to u P such that ∈

1. Tu(P) = Vu Hu is a direct sum; ⊕

2. Hua = (Ra) Hu for each u P and a G, where Ra is the right action on P ∗ ∈ ∈ induced by a G: Rau = ua; ∈

3. Hu depends diﬀerentiably on u.

Hu is called horizontal subspace at u and has the same dimension as the base space M. Vu is called vertical subspace at u and has the same dimension as the structure group G. At this point we look at how the vertical space Vu(P) can be constructed. For an element A of the Lie algebra g of a given Lie group G the right action

Rexp(tA)u = u exp(tA) (3.3) defines a curve through u in P. Since π(u) = π(u exp(tA)) = p, for some p M, 1 ∈ this curve lies within the fibre π− (p). Then we can define the fundamental vector field A TP corresponding to A by ∗ ∈ d A∗ f (u) = f (u exp(tA)) t=0, (3.4) dt | where f : P R is an arbitrary smooth function and u P. Obviously, the −→ ∈ vector A is tangent to V at every u, so that A VuP. We see that the fundamental ∗ ∗ ∈ vector fields associated to generators of the Lie algebra give a basis of the vertical subspace VP. 68 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

3.1.6 Holonomy group

Before going further, we give at this point a brief discussion about holonomy group. The main core of theory is that any kind of connection on a manifold gives rise to some notion of holonomy. The definition for holonomy of connections on principal bundles proceeds in the following fashion. Let G be a Lie group and π : P M a principal G- −→ bundle over a smooth manifold M which is paracompact. Let some connection be defined on P. For a given a piecewise smooth loop γ : [0, 1] M with −→ γ(0) = x for some x M and a point p in the fibre over x, i.e., π(p) = x, the ∈ connection defines a unique horizontal lift γ˜ : [0, 1] P such that −→ d γ˜(0) = p and γ˜(t) H T P. (3.5) dt ∈ γ˜(t) ⊂ γ˜(t) The end point of the horizontal lift over a closed path, will not necessarily be the same point p but rather some other point p g in the fibre over x. Therefore, we · can introduce the notion of a holonomy group for fibre bundles as follows: Let π : P M be a principal G-bundle, p Px, x M. Then the holonomy −→ ∈ ∈ group Holp of P in p is the following subgroup of G:

Holp = g G : loop γ : [0, 1] M γ(0) = p, so that γ˜(1) = p g . (3.6) { ∈ ∃ −→ · } 0 The restricted holonomy group based at p is the subgroup Holp coming from horizontal lifts of contractible loops γ.

3.1.7 Connection one-form

A usual approach of how to separate TuP into VuP and HuP is by introducing a Lie algebra valued one-form ω g T P called the connection one-form. For ∈ ⊗ ∗ each X TuP the value of ω(X) is deﬁned as a unique A g, such that (A )u is ∈ ∈ ∗ the vertical component of X. This projection property can be summarized by the following requirements,

i) ω(A ) = A for any A g; ∗ ∈ 1 1 ii)( Ra) ω = adj(a )ω, i.e., (Ra) ωua(X) = ωua(R (X)) = a ωu(X)a, for all X ∗ − ∗ a∗ − ∈ TuP and a g. ∈ Conversely, for a connection one-form ω satisfying i) and ii) we deﬁne the horizontal subspace HuP as the kernel of ω:

Hu = X TuP ω(X) = 0 . (3.7) { ∈ | } It is clear that the canonical projection π : P M maps the horizontal space −→ Hu(P) onto Tπ(u)M isomorphically. 3.1. YANG-MILLS EQUATION ON MANIFOLD M 69

Let U be an open covering of M and let σ be a local section deﬁned on { α} α each Uα. For each α we deﬁne a Lie algebra valued one-form Aα on Uα, by

Aα = σα∗ ω. (3.8)

The form Aα is also called local connection one-form or gauge potential. In case, a Lie algebra valued one-form Aα on Uα is given, it can be shown that one can uniquely reconstruct the connection one-form ω g Ω1(U ) on π 1(U ) whose ∈ ⊗ α − α pullback by σα is Aα. For an open covering U of M with a family of isomorphisms ψ : π 1(U ) { α} α − α −→ U G and the corresponding family of transition functions ψ : U U G α × αβ α ∩ β −→ we can choose in the principal bundle P a preferred set of local sections σα de- ﬁned via σ (x) = ψ 1(x, e), where x U and e is the identity in G. Then the α α− ∈ α local connection one-forms A = σ ω satisfy on the intersection U U the α α∗ α ∩ β compatibility condition

1 Aβ = adj 1 (Aα) + ψ− dψαβ (3.9) ψαβ− αβ 1 1 = ψαβ− Aαψαβ + ψαβ− dψαβ. (3.10)

3.1.8 Curvature 2-form

In oder to go over to the notion of the curvature of a connection, we recall that a connection ω on a principal bundle P(M, G) separates TuP into HuP VuP. H V H⊕ Accordingly, each vector X of TuP decomposes as X = X + X , where X HuP ∈ and XV VuP. Thus, we can define the concept of covariant derivative on ∈ the principal fibre bundle P as follows. Let φ be a Lie algebra valued r-form, r φ Λ (P) g and let X , ..., Xr TuP be r+1 tangent vectors on P. The (exterior) ∈ ⊗ 1 +1 ∈ covariant derivative of φ is defined via

H H Dφ(X1, ..., Xr+1) = dP(X1 , ..., Xr+1), (3.11)

a a where dPφ = dφ Ta, φ = φ Ta and Ta form a basis of the Lie algebra g. ⊗ ⊗ The deﬁnition of a curvature form now is quite simple. For a given local connection one-form ω on P it is the covariant derivative of it:

Ω = Dω Λ2P g. (3.12) ∈ ⊗ For any curvature form Ω the following theorem called the Cartan structure equation holds:

Theorem 1. Let ω be a connection form and Ω its curvature form. Then 1 dω(X, Y) = [ω, ω](X, Y) + Ω(X, Y), (3.13) −2 for X, Y Tu(P), u P. ∈ ∈ 70 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

The structure equation (3.13) can be expressed as 1 dω = [ω, ω] + Ω, (3.14) −2 or dω = ω ω + Ω (3.15) − ∧ Here we use the Lie bracket deﬁned for two Lie algebra valued forms α Λp g ∈ ⊗ and β Λq g as ∈ ⊗ [α, β] = α β ( 1)pqβ α (3.16) ∧ − − ∧ a b pq b a = TaT α β ( 1) T Taβ α (3.17) b ∧ − − b ∧ a b c a b = [Ta, T ]α β = f Tcα β , (3.18) b ∧ ab ∧ c g where fab are structure constants of the Lie algebra and Ta are its generators. Furthermore, it can be shown that every curvature form Ω fulﬁlls the Bianchi identity: DΩ = 0.

3.1.9 Bianchi equation

Analogously to the definition of the local connection one-form Aα on M we use a local section σα to pull back a global curvature form Ω on P to a local object Fα living on a patch of M. Namely, for a given section σ on U M we define the α α ⊂ local (Yang-Mills) field strength by

2 F = σ∗ Ω Λ (U ) g. (3.19) α α ∈ α ⊗ One can show that F is formally expressed in terms of the gauge potential as

F = dA + A A . (3.20) α α α ∧ α µ Locally we can write the connection and curvature forms as A = Aµdx and F = 1 F dxµ dxν (here we omit the subscript α). Then we can ﬁnd that the 2 µν ∧ components of F are related to those of A in the following way,

F = ∂ A ∂ A + [A , A ]. (3.21) µν µ ν − ν µ µ ν With this relation one can obtain that on intersection of two local trivializations U , φ and U , φ the transformation of the curvature form has a simply form { α α} { β β} 1 Fβ = gαβ− Fα gαβ, (3.22)

2 where gαβ is the corresponding transition function. Moreover, since d = 0, one ﬁnds that

dF = dA A A dA = F A A F (3.23) α α ∧ α − α ∧ α α ∧ α − α ∧ α = [A , F ]. (3.24) − α α 3.2. YANG-MILLS EQUATIONS ON G/H 71

This equation results in the local identity called the Bianchi equation:

F = ddF + [A , F ] = 0, (3.25) Dα α α α α where the covariant derivative is deﬁned by D = d + [A , ]. (3.26) Dα α ·

3.2 Yang-Mills equations on G/H

We have already gathered all needed ingredients to start our consideration of the Yang-Mills theory on coset spaces in order to ﬁnally go over to the concrete study of its equations on nearly Kahler¨ manifolds. We begin with the most important things and end up with some solutions of the Yang-Mills equations to the issue of the chapter.

3.2.1 Coset space G/H In the following we want to concretize our consideration supposing that M is a reductive homogeneous space of the form G/H, where G is a Lie group and H is its closed subgroup. Let IA , A = 1, ..., dim G, be the generators of the Lie group C { } G and fAB the corresponding structure constants,

C [IA, IB] = fABIC. (3.27)

Since the Lie algebra g of G can be decomposed as g = h m, where h is the Lie ⊕ algebra of H and m is the orthogonal complement of h in g, as we have already seen in section 2.7.4 the generators IA split in two subsets Ii , i = 1, ..., dim H, { } { } and Ia , a = 1, ..., dim G dim H, of the generators of h and m, respectively. As { } − usual we identify the subspace m of g with the tangent space of G/H at any given point x G/H. The commutator relations for reductive homogeneous space ∈ satisfy (see equations (2.172))

k c i c [Ii, Ij] = fijIk, [Ii, Ia] = fiaIc, [Ia, Ib] = fabIi + fabIc. (3.28)

The Killing-Cartan metric on the Lie algebra g can be taken to have the form

C D gAB = fAD fCB = δAB (3.29) after an appropriate normalization of the generators IA. Explicitly one obtains

k l b a gij = fil fkj + fia fbj = δij, gia = 0, (3.30) i d c d gab = 2 fad fib + fad fcb = δab. (3.31)

On the coset space G/H we use the induced metric gab coming from gAB on G. 72 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

3.2.2 Connection on G/H

The basis elements IA of the Lie algebra g can be represented by left-invariant A vector fields EÂ on the Lie group G. We denote by eˆ the set of left-invariant one-forms dual to EÂ. The space G/H consists of left cosets gH and the natural projection π : G G/H, g gH. Over a small contractible open subset U of → 7→ G/H, one can choose a map L : U G such that π L is the identity, i.e., L is → ◦ a local section of the principal bundle G G/H. The pullbacks of eÂ by L we A a → denote e . In particular, e form an orthonormal frame for T∗(G/H) over U, so i i a i we can write e = eae with real functions ea. The dual frame for T(G/H) will be denoted by Ea. For a given connection 1-form on G/H, which is eventually torsion-full, the following equation holds (compare with (2.7.6):

dea + ωa eb = Ta, (3.32) b ∧ where Ta = 1 Ta eb ec. The application examples, which are the most interesting 2 bc ∧ for us, allow to choose the torsion components proportional to the structure constants,

a a Tbc = κ fbc, (3.33) where κ is an arbitrary real parameter. If we look from the other point of view, we can use the Mauer-Cartan equation A A to get similar equation. Namely, the 1-forms θ := e IA satisfy ⊗ dθ = θ θ. (3.34) − ∧ This leads to the relations 1 1 1 dea = f a ei eb f a eb ec, dei = f i eb ec f i ej ek. (3.35) − ib ∧ − 2 bc ∧ −2 bc ∧ − 2 jk ∧ Plugging (3.32), (3.33) and (3.35) together we ﬁnd that [70, 71]

1 ωa = f a ei + (κ + 1) f a ec =: ωa ec, (3.36) b ib 2 cb cb where we deﬁne 1 ωa = f a ei + (κ + 1) f a . (3.37) cb ib c 2 cb

3.2.3 Yang-Mills equations

Consider the principal ﬁbre bundle P(G/H, G) over G/H with the structure group G. As we learned from the previous section a g-valued connection, one-form A 3.2. YANG-MILLS EQUATIONS ON G/H 73 on P and the corresponding ﬁeld strength = d + , which can locally F A A ∧ A be expressed in the basis ea as { }

a 1 a b = Aae and = F e e , (3.38) A F 2 ab ∧ fulﬁll the Yang-Mills equation. In components this equation is given by

ab ab ab aF = aF + [Aa, F ] (3.39) D ∇ ab d ab b ad ab = EaF + ωdaF + ωadF + [Aa, F ] = 0. (3.40)

Note that since we work on a non-holonomic basis Ea , we must use at this { } point the covariant derivative associated to the torsion-full connection ω on ∇ G/H from (3.32) (see [70, 71] and references therein).

3.2.4 Hermitian-Yang-Mills equations

In this section we discuss a special class of solutions to the Yang-Mills equations (3.25). This class includes Hermitian-Yang-Mills connections on Kahler¨ manifolds. Let π : E M2m be a unitary bundle of complex rank r, i.e., a vector bundle −→ arises from a unitary representation of a structure group G (the simplest case is of cause G =U(r), so that E = M U(r)). Let M be a complex m-dimensional × Kahler¨ manifold with local real coordinates x = (xi) and the tangent space basis ∂ i j m g ∂i := ∂xi for , = 1, ..., 2 . Then a metric and the corresponding Kahler¨ two- 2 i j i j form ω read ds = gijdx dx and ω = ωijdx dx , respectively. We consider on ∧ ∧ the complex vector bundle E a gauge connection and the curvature two-form A = d + corresponding to . For any connection on E over M the F A A ∧ A A A components Ai and Fij take values in u(r). Moreover, the 2-form decomposes F as

= 2,0 + 1,1 + 0,2, (3.41) F F F F where 2,0 denotes the (2, 0)-part of ( 0,2 = ( 2,0) ), and 1,1 denotes F F F − F ∗ F the (1, 1)-part of with respect to the complex coordinates: zi = xi + ixm+i, F z¯i = xi ixm+i, i = 1, ..., m. We call a Hermitian-Yang-Mills connection on E if − A is unitary and A 1,1 ω = λId, 0,2 = 0, (3.42) F · F m 1 m(C1(E)[ω] − where λ = r[ω]m is a constant factor, C1(E) the ﬁrst Chern class of E and [ω] denotes the cohomology class represented by ω. In particullar, after simple changing of the ﬁeld strength by performing the shift 1 tr( ) F F −→ F − rankE F one can assume that λ = 0 [72]. Equations (3.42) are known as Donaldson- Uhlenbeck-Yau or Hermitian-Yang-Mills equations [73, 74, 75]. In the special 74 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS case of an almost complex 4-manifold with a metric g they coincide with the anti-self-dual Yang-Mills (ASDYM) equations

= , (3.43) ∗ F −F where is the Hodge operator. ∗ 3.2.5 Some explicit solutions of the Hermitian-Yang-Mills equations

Solutions of the HYM equation on CPn

In [75] it was shown that some explicit solutions of the Hermitian-Yang-Mills equations on CPn can be obtained in the commutative limit from non-commutative instanton-type solutions on CPn. We briefly sketch the ansatz and the construction for the commutative case. We consider the group U(n + 1). The complex projective space CPn can be considered as the quotient space U(n + 1)/U(1) U(n) with the projection × U(1) U(n) U(n + 1) × CPn (3.44) −−−−−−−→ and fibres U(1) U(n). For g U(n + 1) the canonical one-form Ω = g dg on × ∈ † U(n + 1) takes values in the Lie algebra u(n + 1) and satisfies the Maurer-Cartan equation. We define a (n + 1) (n + 1) matrix V U(n + 1) as the matrix × ∈ 1 1 ! γ− γ− Y† V = 1 − , (3.45) Yγ− Λ where  1   y     .  Y :=  .  (3.46)    yn  is a complex vector, p γ := ya y¯a + 1 (3.47) and 1 Λ = Idn Y Y†. (3.48) − γ(γ + 1)

With this construction we can build a connection 1-form

= V†dV (3.49) A with the vanishing curvature two-form = + = 0. F A A ∧ A 3.2. YANG-MILLS EQUATIONS ON G/H 75

The ﬂat connection form can be rewritten in a block form A ! a φ = − † (3.50) A φ A with a u(1) and A u(n). The values of φ, a and A can be obtained from the ∈ ∈ deﬁnition of (3.49). Finally, one founds that the two-forms A a¯ b Fu = da + a a = φ† φ = δ φ¯ φ , (3.51) (1) ∧ ∧ ab¯ ∧ a b¯ Fu = dA + A A = φ φ† = (φ φ¯ ) (3.52) (n) ∧ ∧ ∧ ¯ obey the Hermitian-Yang-Mills equation with respect to the basis φa φ¯ b of { ∧ } (1, 1)-forms:

Fab = 0 = Fa¯b¯, Faa¯ = F11¯ + ... + Fnn¯ = Idn (3.53) and

fab = 0 = fa¯b¯, faa¯ = f11¯ + ... + fnn¯ = n, (3.54) where Fab, Faa¯ and fab, faa¯ are components of Fu(n) and Fu(1), respectively. The matrix V from (3.45) defines an embedding of CPn into U(n + 1). For this embedding the one-form Ω on CPn matches with the flat connection on A U(n + 1) given by (3.49). The splitting of in components a, A, φ corresponds to A the decomposition of in one-forms on CPn with values in u(1) u(n) and into A n ⊕ one-forms lying in T(CP ). The gauge potential A defining Fu(n) and satisfying the Hermitian-Yang-Mills equation coincides with the canonical connection on CPn.

Anti-self-dual gauge ﬁelds on M4 = S4

As we have already mentioned, on four dimensional manifolds the DUY equations are equivalent to the anti-self-dual Yang-Mills equation = . We ∗F −F give here an explicit solution of the ASD for the special example M4 = S4 = Sp(2)/Sp(1) Sp(1). Let start with a ﬂat connection on the trivial vector bundle × A M4 C4 M4 given by the one-form × −→ ! 1 A− φ = Q− dQ =: −+ , (3.55) A φ† A where Q’s are 4 4 matrices in Sp(2) SU(4). We consider the local sections Q of × ⊂ the form ! ! 1 Id2 x 1 1 Id2 x Q := f − 2 − and Q− = Q† = f − 2 , (3.56) x Id2 x Id2 † − † where

µ µ µ ν x = x τµ, x† = x τµ† , f := 1 + x†x = 1 + δµνx x , (3.57) 76 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS and

i i (τ ) = ( iσ , Id2) and (τ† ) = (iσ , Id2) (3.58) µ − µ with

τµ† τν =: δµνId2 + ηµν (3.59) Here xµ are local coordinates on U S4. From (3.55) and (3.56) we obtain ⊂ 1 µ ν A− = η¯ x dx su(2), (3.60) f µν ∈ 1 A+ = η xµdxν su(2), (3.61) f µν ∈ ! 1 i dx3 + idx4 dx1 idx2 φ = dx = − − = (3.62) f f dx1 + idx2 (dx3 idx4) ! − − ! i dz dy¯ i θ2 θ1¯ = − =: − , (3.63) f dy dz¯ 2Λ θ1 θ2¯ − − with 2Λdy 2Λdz θ1 := = (θ1¯ ), θ2 := = (θ2¯ ), (3.64) 1 + r2 1 + r2 where the bar denotes complex conjugation and the real parameter Λ is the radius of S4. The connection (3.55) is ﬂat, so we get

= d + (3.65) F A A ∧ A ! F φ φ (dφ + A φ + φ A+) = − − ∧ † − − ∧ ∧ (3.66) dφ + A+ φ + φ A F+ φ φ † ∧ † † ∧ − − † ∧ = 0, (3.67) where F = dA + A A (3.65) implies ± ± ± ∧ ± ! 1 θ1 θ1¯ θ2 θ2¯ 2θ1¯ θ2 F− = φ φ† = ∧ − ∧¯ ¯ ∧ ¯ , (3.68) ∧ 4Λ2 2θ1 θ2 θ1 θ1 + θ2 θ2 − ∧ − ∧ ∧ 1 1¯ 2 2¯ 1¯ 2¯ ! + 1 θ θ + θ θ 2θ θ F = φ† φ = ∧ ∧ ¯ ∧ ¯ . (3.69) ∧ 4Λ2 2θ1 θ2 θ1 θ1 θ2 θ2 − ∧ − ∧ − ∧ It can easily be shown that F− is anti-self-dual (ASD) gauge ﬁeld on a rank-2 complex vector bundle E M4, i.e., F = F . This also means that F obeys −→ ∗ − − − − the HYM equation as well. These solution of ASD equations on S4 can be used in further applications for instance for ﬁnding some solutions of HYM equations on twistor space of S4. This is a special sort of spaces which is in case of S4 in particular Kahler¨ and homogeneous. The complete information about solutions to HYM on S4 and some other four dimensional manifolds as well as their twistor spaces can be found in reference [72]. 3.3. MANIFOLDS WITH G2-STRUCTURE 77

Solutions of HYM equations on Calabi-Yau cones

We do not give here the explicit solutions of HYM equation on Calabi-Yau cones, but refer the reader to the paper of Correia [76], where the author approaches 2dC-dimensional Calabi-Yau cones and suggests explicit solutions of the ﬂow equations that correspond to non-trivial SU(dC) HYM instantons.

3.3 Manifolds with G2-structure

At first we are going to introduce the notion of a G2-structure and in this context take a look at the compact 6-dimensional nearly Kahler¨ coset spaces G/H. Such manifolds are important examples of SU(3)-structure manifolds used in flux compactifications of string theories.

3.3.1 Seven-dimensional manifolds with G2-structure

In this section we give a brief introduction to G2 geometry, following [63]. − Beginning with a 7-dimensional vector space V7, we consider the group GL(7, R) 7 1 acting on Λ3(V ) with two open orbits. For v in one of these orbits the stabilizer St(v) GL(7, R) is 14-dimensional. It can easily be shown that St(v) is a real ∈ representation of a Lie group G2. Depending on in which orbit v is, the stabilizer is either compact or non-compact form of G2. A 7-manifold M equipped with 7 a 3-form φ Λ3(V ) is called a G2-manifold if φ is stable everywhere in M. In ∈ this case we say that the structure group of M is reduced to G2. Moreover, on M there can be declared a natural Riemannian structure gφ: Z x, y (φyx) (φyy) φ, x, y TM. (3.70) −→ ∧ ∧ ∀ ∈ M

One says that G2-manifold is parallel if φ = 0, where is the Levi-Civita ∇ ∇ connection associated with this Riemannian structure. In local coordinates x1, ..., x7 of R7 the 3-form φ can be given by { } φ = dx123 + dx145 + dx167 + dx246 dx257 dx347 dx356, (3.71) − − − 123 1 2 3 7 with exterior form dx = dx dx dx on R . In each case a G2-structure ∧ ∧ fixes the 4-form φ, the Euclidean metric g = (dx1)2 + ...(dx7)2 and orientation of ∗ M [77]. A G2-manifold may contain isolated singularities whose simplest examples are conical singularities. In general, a metric space X with fixed points x1, ..., xn is called a space with isolated singularities if X x1, ..., xn is a Riemannian manifold. \{ } 1 For every x in a set X and a group G acting on X the stabilizer subgroup of X is defined as the set Gx of all elements of X fixing x, Gx g G gx = x . { ∈ | } 78 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

Suppose, there is only one singular point x X. Then the singularity is called ∈ conical if X is equipped with a ﬂow acting on X by homotheties and contracting X to x. In this case, X is isomorphic to a Riemannian cone of a Riemannian manifold M. As it will be discussed in the next section, a cone C(M) of a nearly Kahler¨ manifold is always equipped with a parallel G2-structure. The converse statement is also true: every conical singularity of a parallel G2-manifold is obtained as C(M), for some nearly Kahler¨ manifold M (see for example [63]). In general there exist exactly 16 classes of G2-structures. For more information see for instance [78]. Moreover, we remark that G2 manifold is always Ricci-ﬂat, and a G2 structure φ is torsion-free if and only if it is both closed and coclosed: dφ = 0 and dψ = d( g ρ) = 0 [79]. ∗ φ 3.3.2 Cones over nearly Kahler¨ 6-manifolds

First of all we repeat what we have already learned about nearly Kahler¨ manifolds in previous chapter. An almost Hermitian manifold (M, g, J), with fundamental two-form ω( , ) = g(J , ) and Levi-Civita connection is called nearly · · · · ∇ Kahler¨ if the corresponding canonical Hermitian connection ¯ has a totally skew- ∇ symmetric torsion. It is also known that nearly Kahler¨ manifolds with integrable almost complex structure are necessarily Kahler.¨ The non-trivial case is then the so-called strict nearly Kahler¨ manifold, when dω = ω = 3 J , 0 at every point ∇ ∇ of M. It was shown by A. Gray and P.A. Nagy (see [80] and references therein) that any nearly Kahler¨ manifold is locally a product of 6-dimensional strict nearly Kahler¨ manifolds, locally homogeneous manifolds and twistor spaces of positive quaternionic Kahler¨ manifolds. Due to this result, the most interesting objects remaining to be studied are strict nearly Kahler¨ structures in dimension 6. We remark that 4-dimensional nearly Kahler¨ manifolds are automatically Kahler.¨ On the other hand, the dimension 6 is particularly important because of some amount of properties which we discuss in this section. One of the important results is the fact that nearly Kahler¨ structures are closely related to G2 structures via the cone construction. To see this at ﬁrst we give a description of the kind of objects cones are.

Deﬁnition 5. Let (M, g) be a Riemannian manifold. The Riemannian cone of M is

2 2 C(M):= (M R 0, t g + dt ), × > where t denotes the coordinate on the half-line R>0.

Depending on the cone structure of a manifold M one can make some conclu- sions about the geometry of the basis manifold M. If C(M) has special holonomy, then the geometry of M can be given in the following table. Furthermore, im- 3.3. MANIFOLDS WITH G2-STRUCTURE 79

Table 3.1: Riemannian cones with special holonomy Holonomy of C(M) Geometry of C(M) Geometry of M SO(n) Riemannian - U(n) Kahler¨ Sasakian SU(n) Calabi-Yau Sasaki-Einstein Sp(n) hyper-Kahler¨ 3-Sasakian Sp(n)Sp(1) quaternionic-Kahler¨ - G2 G2-manifolds nearly Kahler¨ Spin(7) Spin(7)-manifolds nearly G2-manifolds portant statements [80, 81] are:

A Riemannian manifold (M6, g) carries a nearly Kahler¨ structure if and • 6 2 2 only if its cone (M R 0, t g + dt ) has holonomy contained in G2. × > Real Killing spinors on M correspond uniquely to parallel spinors on C(M). • This implies that then C(M) is Ricci-ﬂat. G2-manifolds also belongs to the Ricci-ﬂat manifolds and as follows admit parallel spinors.

More information can for instance be found in [63]. As an example of cone over nearly Kahler¨ manifold we mention R7 which is the cone over the sphere S6.

3.3.3 Special G -manifolds of type R G/H with nearly Kahler¨ coset 2 × spaces G/H

In this section we consider a special type of G2-manifolds, namely the cylinder spaces R G/H over nearly Kahler¨ coset spaces G/H. So suppose that the × coset space G/H is nearly Kahler.¨ This means that the manifold G/H admits a Riemannian metric g, an almost complex structure J, a real two-form ω and a complex three-form Ω reducing the structure group of the tangent bundle to SU(3) and satisfying the compatibility condition g(J , ) = ω( , ). With respect · · · · to J, the forms ω and Ω are of type (1, 1) and (3, 0), respectively. We ﬁx their normalization by the volume form Vg of g so that

ω ω ω = 6Vg and Ω Ω¯ = 8iVg. (3.72) ∧ ∧ ∧ − This SU(3)-structure on the coset space is nearly Kahler¨ if additionally

dω = 3ρImΩ and dΩ = 2ρω ω (3.73) ∧ for some real non-zero constant ρ (in case ρ = 0 the manifold is Calabi-Yau). Moreover, an important property of the compact nearly Kahler¨ manifolds in general is a 3-symmetry. It manifests itself in the existence of an automorphism 80 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS s on the structure group G with the features s3 =Id and s(H) = H. This automorphism is special, because it gives rise to definition of a canonical almost complex structure on the coset space itself. This proceeds as follows. Since s acts on G/H, there is a corresponding automorphism S : g g on the −→ Lie algebra g = h m of G, which in particular is trivial on the h, the Lie algebra ⊕ of the fixed subgroup H G. It can be shown that on m the map S defines an ⊂ almost complex structure J : m m via −→ 1 √3 2π S m = + J = exp( J) (3.74) | −2 2 3 where J satisfies J2 = Id. Therefore, the G-invariant metric g, the almost − complex structure and the nearly Kahler¨ form ω on the coset G/H have in terms a of the local orthogonal frame e and the corresponding dual frame Ea the { } following form

1 g = δ eaeb, J = JbeaE and ω = J ea eb. (3.75) ab a b 2 ab ∧ The Mauerer-Cartan equations (3.35) imply for dω and dω that ∗ 1 1 dω = J (dea eb ea deb) = f˜ ea eb ec, (3.76) 2 ab ∧ − ∧ −2 abc ∧ ∧ 1 dω = f ea eb ec, (3.77) ∗ 2 abc ∧ ∧ where

f˜abc := fabd Jdc (3.78) are components of a totally antisymmetric tensor on G/H. Because of (3.73), we see that ImΩ should be normalized so that ImΩ 2 = (ImΩ) (ImΩ) = || || abc abc 4. Besides, on any nearly Kahler¨ coset the following identities between the components of J and the structure constants hold [2]

1 f f = f f = δ , (3.79) aci bci acd bcd 3 ab Jcd fadi = Jad fcdi and Jab fabi = 0. (3.80)

Hence, from (3.79) we obtain

ω 2 = 3, (3.81) || || and consequently

1 a b c 1 a b c 1 ImΩ = f˜abce e e , ReΩ = fabce e e and ρ = . (3.82) − √3 ∧ ∧ − √3 ∧ ∧ 2 √3 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 81 2 ×

Now we show that R G/H can easily be furnished with a G2-structure. All × what we need is already there. We use the forms deﬁned above to introduce the three-forms Σ = e0 ω + ImΩ (3.83) ∧ and 0 Σ0 = e ω + ReΩ. (3.84) ∧ Both three-forms (3.83) and (3.84) deﬁne a G2-structure on R G/H, i.e., a reduc- × tion of the holonomy group SO(7) to a subgroup G2 SO(7). Moreover, rrom ⊂ (3.83) and (3.84) one sees that both G2-structures are induced by SU(3)-structure of G/H.

3.4 Yang-Mills on G -manifolds R G/H 2 × After a long preparation we now are able to apply the collected toolkits and try to search for solutions of the Yang-Mills theory on the nearly Kahler¨ coset spaces explicitly. Although the finding of a general solution is highly intricate, it will be possible give some specific solutions, in particular those which follow from certain flow equations.

3.4.1 Yang-Mills equations on spaces R G/H: G-invariant ansatze¨ × and reductions to kink-type equations

On a d-dimensional Riemannian manifold, the Yang-Mills equation with torsion has the form

+ = 0, (3.85) D ∗ F ∗H ∧ F where is a three-form. In this section we consider this equation (3.85) on the H special case of seven-dimensional manifolds R G/H, with G/H a nearly Kahler¨ × coset space (see [2]). For the metric and the volume form we choose

2 g7 = (e0) + g6 and V7 = e0 V6, (3.86) ∧ where e0 = dτ and τ is a coordinate on R, while g6 and V6 are metric and volume form on G/H. Since can be chosen freely, we suppose that H 1 = κdτ dω, (3.87) ∗ H −3 ∧ with real constant κ. 0 a For the connection one-form = 0e + ae we make the most general A A A G-invariant ansatz,

i a = e Ii + e Xa(τ), with Xa(τ) = λaIa + µa J I , (3.88) A ab b 82 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

where λa and µa are real functions of τ. We remark that the gauge freedom allows us to set here 0 = 0 without any lost of generality. A 0 a 1 a b Computation of the curvature = d + = F0ae e + F e e of F A A ∧ A ∧ 2 ab ∧ this connection gives in components

i c i c F = ( f Ii + f Xa [Xa, X ]) + e ([Ii, Xa] f Xc) and F0a = X˙ a, (3.89) ab − ab ab − b b − ia ˙ ∂Xa with Xa = ∂τ . Due to the G-invariance the last term of Fab in (3.89) vanishes and

c [Ii, Xa] = fiaXc. (3.90)

In order to write the Yang-Mills equation in components, it is necessary to introduce the torsionful spin connection on G/H. Recall that a linear connection a c a c is a matrix of one-forms Γb = e Γcb. The connection is metric compatible if Γa gcb is anti-symmetric. Its torsion Ta = 1 Ta eb c is given by the structure equation 2 bc ∧ dea + Γa e = Ta. (3.91) b ∧ b Choosing Ta = eay Ta = κ f a , (3.92) − H ⇐⇒ bc bc the unique metric-compatible linear connection with this torsion reduces to

1 Γa = ei f a + (κ + 1) f a . (3.93) cb c ib 2 cb The torsionful spin connection on R G/H is then given by Γa , with additional × cb condition

0 a 0 Γ0b = Γ0b = Γcb = 0. (3.94)

Putting (3.91) in the Yang-Mills equation with torsion (3.85) one gets

0b ab d ab b cd ab E0F + EaF + ΓdaF + ΓcdF + [Aa, F ] = 0, (3.95) a0 d a0 a0 EaF + ΓdaF + [Aa, F ] = 0. (3.96)

We are now going to evaluate these Yang-Mills equation in terms of our ansatz (3.88). To do this we need to take in consideration the structure constant identities coming from the Jacobi-identities between the generators IA (see (A.3)):

i c i a i j d c d a d b fba fkb + fcd fkd + fkj fca = 0, fba fkb + fcb fkb + fkb fca = 0, (3.97) and that they additionally satisfy

1 1 f i f a = 0, f a f b = δab, f k f d = δd. (3.98) cd cd cd cd 3 ba kb 3 a 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 83 2 ×

These relations will enormously simplify the outgoing expressions. One more c point is that the tensor fab is totally antisymmetric. This implies that for the following arising term vanishes:

k d k a i c i c e f [X , Xa] e f [Xc, X ] e f [X , Xc] + e f [Xa, Xc]) = d kb b − c kd d − b ia b b ib k d k b k a k b = (ed fkb + ec fkc)[Xb, Xa] + (ed fkb + ec fka)[Xc, Xb] = 0 (3.99) Plugging now the ansatz (3.88) into (3.95), (3.96) we obtain at ﬁrst

i d j c X¨ a = e f ( f Ij + f Xc [X , Xa]) d ib ba ba − b k a 1 a i b +(e f + (κ + 1) f )( f Ii + f X [Xc, X ]) c kd 2 cd cd cd b − d i k d +[e Ii + X , f I + f X [X , Xa]] (3.100) b b ba k ba d − b i c i a i j k d c d a d b k = ( fba fkb + fcd fkd + fkj fca)ecIi + ( fba fkb + fcb fkb + fkb fca)ecXd 1 1 + f i f a (κ + 1)I + f a f b (κ + 1)X + f k f d X 2 cd cd i 2 cd cd b ba kb d k d k a i c i c (e f [X , Xa] e f [Xc, X ] e f [X , Xc] + e f [Xa, X ]) − d kb b − c kd d − b ia b c ib b 1 a d (κ + 1) f [Xc, X ] + f [X , X ] [X [X , Xa]]. (3.101) −2 cd d ba b d − b b Next we take (3.98) and (3.99) into account, pull down the indices and ﬁnd that the Yang-Mills equations can be written in terms of Xa as κ 1 κ + 3 X¨ a = − Xa f [Xc, X ] [X , [X , Xa]] (3.102) 6 − 2 acd d − b b and

[Xa, X˙ a] = 0, (3.103) respectively.

3.4.2 General solution of the G-invariance condition

In this section we consider how the G-invariance condition

b [Ii, Xa] = fiaXb (3.104) inﬂuences solutions Xa of the Yang-Mills equations (3.102) and (3.103), i.e., how it determines a speciﬁc form of the solutions. First of all (3.88) and (3.104) directly imply

Xab fbci = fiabXbc. (3.105)

The general solution of (3.104) reads

(p) (p) Z = φpI , with φp C; p = 1, ..., q; αp = 1, ..., dp, (3.106) αp αp ∈ 84 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

Pq (p) where dp = 3 and I are complex linear combinations of Ia such that p=1 { αp } { }

(p) βp (p) [Ii, I ] = f I . (3.107) αp iαp βp

p Notice that for diﬀerent p each group of I( ) is closed for itself. The regulation { αp } of the grouping to diﬀerent p is done according to decomposition of the adjoint representation of the gauge group G into the irreducible adjoint representation H and the rest, Xq Xq adj(G) = adj(H) Rp R¯ p. (3.108) ⊕ ⊕ p=1 p=1

For all nearly Kahler¨ cases we are interested in representations Rp are complex. Due to the fact that both (p) (p) ¯ (p) ¯ ¯(p) Zαp = φpIαp and Zαp = φpIαp (3.109) are solutions of (3.104), for each value of (p, αp) the antihermitian solutions Xa can be constructed as

(1) 1 (p) (p) (p) (p) X = (Z + Z¯ ) = ReφpReI ImφpImI , (3.110) αp 2 αp αp αp − αp (2) 1 (p) (p) (p) (p) X = (Z Z¯ ) = ImφpReI + ReφpImI . (3.111) αp 2i αp − αp αp αp For further computation, such as in calculations of the potential, we will need the trace trR(IaIb) over the chosen representation R of G. Although the gauge group G can in general be represented arbitrarily, in order to work with and taking values in the adjoint representation of G, we should chose the A F smallest non-trivial matrix representation of G to describe the generators Ii, Ia and solutions Xa. In the following we devote ourself to consideration of relevant cases.

G2/SU(3)

The smallest non-trivial matrix representation 14(G2) of the group G2 decomposes in

14(G2) = 8 3(SU(3)) 3(SU(3))¯ , (3.112) adj ⊕ ⊕ (1) (1) (1) ¯(1) ¯(1) ¯(1) so that q = 1 and d1 = 3. I1 , I2 , I3 and I1 , I2 , I3 generate 3(SU(3)) and 3(SU(3)),¯ respectively, and are linear combinations of I , ...., I6 . Since there is { 1 } only one group, q = 1, we have only one complex parameter φ:

(1) (1) Zα1 = φIα1 , with α1 = 1, 2, 3. (3.113) As an appropriate matrix representation one may take the 7-dimensional fundamental of G2. This case was already treated in [2]. 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 85 2 ×

SU(3)/U(1) U(1) × In this case we have

8(SU(3)) = ((0, 0) (0, 0)) (2, 0) ( 2, 0) ⊕ adj ⊕ ⊕ − ⊕ (1, 3) ( 1, 3) ( 1, 3) (1, 3), (3.114) ⊕ ⊕ − − ⊕ − ⊕ − (1) (1) (2) (2) (3) (3) where the corresponding generators are I7, I8, I , I¯ , I , I¯ , I , I¯ (notice that there is no index αp, because of the representation of H is one-dimensional) and the notation is according to: (q7, q8) has under the action of (I7, I8) the charge 1 1 ( − q7, − q8). Since q = 3, there are three complex parameter φ1, φ2, φ3 and 2 √3 6

(p) (p) (p) (p) Z = φpI , Z¯ = φ¯ pI¯ , with p = 1, 2, 3. (3.115)

Using the structure constants given in [82] we furnish the generators IA with an additional factor 1 and respectively get f A 1 f A (see also 3.159-3.160). √3 CD −→ √3 CD Then due to i [I7, I1 + iI2] = − (I1 + iI2), [I8, I1 + iI2] = 0, (3.116) √3 i i [I7, I3 + iI4] = − (I3 + iI4), [I8, I3 + iI4] = − (I3 + iI4), (3.117) √3 2 i i [I7, I5 + iI6] = (I5 + iI6), [I8, I5 + iI6] = − (I5 + iI6), (3.118) √3 2 one obtains

(1) (2) (3) I = I1 + iI2, I = I3 + iI4, I = I5 + iI6. (3.119)

Then we have 1 X = (Z(1) + Z¯ (1)) = Re(φ )I + Im(φ )I , (3.120) 1 2 1 1 1 2 1 (1) (1) X3 = (Z Z¯ ) = Im(φ )I Re(φ )I2, (3.121) 2i − 1 1 − 1 and analogously for X3, X4 and X5, X6.

Sp(2)/Sp(1) U(1) × Here we have the following decomposition of the group Sp(2),

10(Sp(2)) = (30 10)adj 21 2 1 12 1 2, (3.122) ⊕ ⊕ ⊕ − ⊕ ⊕ − where in the designation of nq the number n denotes the dimension of the corresponding irreducible part in the representation and q is the double charge under the action of U(1). Note that n¯ q = n q. The corresponding generators − 86 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

(1) (1) (2) are I8, I9, I and I7 of (30 10) , I , I of (2 ) and I of 12 (and the complex 10 ⊕ adj 1 2 1 conjugate generators for 2 1, 1 2, respectively). Since q = 2, we get two parameter − − φ1 and φ2, so that

(1) (1) (2) (2) Z = φ1I , Z = φ2I , and c.c. (3.123)

Choosing the structure constants used in [82] and rescaling the generators IA with a factor 1 , we have to rescale f A 1 f A (see also section 3.4.5). Then √3 BC −→ √3 BC we get (notation: adj(A) B = [A, B]) ·

adj(I8, I9, I 0) (α(I + iI2) + β(I3 + iI )) = γ(I + iI2) + δ(I3 + iI ) (3.124) 1 · 1 4 1 4 { n = 2

adj (I8, I9, I 0)(I5 + iI6) = 0 { n = 1 (3.125) · 1 i [I , I + iI ] = (I + iI ) { q = 1, (3.126) 7 1 2 2 1 2 i [I , I + iI ] = (I + iI ) { q = 1, (3.127) 7 3 4 2 3 4 [I7, I5 + iI6] = i(I5 + iI6) { q = 2. (3.128)

Consequently,

(1) (2) (3) I1 = I1 + iI2, I2 = I3 + iI4, I = I5 + iI6, and c.c. (3.129) Then solutions of the G-invariance condition for corresponding Z(i) are

1 (1) (1) 1 (1) (1) X = (Z + Z¯ ) X2 = (Z Z¯ ), (3.130) 1 2 1 1 2i 1 − 1 1 (1) (1) 1 (1) (1) X3 = (Z + Z¯ ) X = (Z Z¯ ), (3.131) 2 2 2 4 2i 2 − 2 1 (2) (2) 1 (2) (2) X5 = (Z + Z¯ ) X6 = (Z Z¯ ). (3.132) 2 2i − (3.133)

3.4.3 Calculation of traces over IAIB

At ﬁrst, we consider the traces trR in an arbitrary representations R of a simple group G. Let TA, A = 1, ..., d = dim(G), be the generators of G and κAB the Killing form. Then,

trRTATB = χRκAB, (3.134) where χR is known as the (2nd-order) Dynkin index of R. Any rescaling of the TA changes the structure constants and κAB, but the ratio to the adjoint representation R = adj (as a reference) does not change. Therefore,

χR trRTATB = tradjTATB. (3.135) χadj 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 87 2 ×

The Dynkin index is related to the value of the quadratic Casimir in the given representation (of dimension dR):

dadjχR = dRCR, (3.136) where CR is deﬁned via X AB κ TATB = CRIdR (3.137) A,B

AB and κ is the inverse of κAB. In this thesis we choose the following normalization

D C tr TATB = f f = δAB. (3.138) adj AC BD − Hence, X 2 D B B (TA) = ( f f ) = (δ ). (3.139) AC AD − C A

Therefore, comparing (3.134) and (3.138), we ﬁnd that κAB is proportional to δAB, 1 and ratios CR/C of quadratic Casimirs do not change if we replace κ by δAB adj AB− − in (3.137), i.e., X 2 (TA) = C0 (Id)R. (3.140) − R A

In particular, Cadj0 = 1. Therefore, combining (3.135) and (3.136) we get (set dadj = d)

dRCR dRCR0 dR trRTATB = δAB = δAB = CR0 δAB −dCadj −dCadj0 − d

=: χ0 δAB, (3.141) − R deﬁning the proportionality

dR trR χR0 = CR0 = . (3.142) d tradj which is independent of normalization. In particular, χadj0 = 1. It remains to compute CR0 from (3.142).

Case SU(3)

The 3-dimensional representation R = ‘30 (see (3.4.4)) yields 4 4 3 1 C0 = , d3 = 3, d = 8 = χ0 = = , (3.143) 3 9 ⇒ 3 9 8 6 1 thus, tr3 = 6 tr8 for G =SU(3). 88 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

Case Sp(2)

In the 4-dimensional representation R = ‘40 (3.4.5) we get 5 5 4 1 C0 = , d = 4, d = 10 = χ0 = = , (3.144) 4 12 4 ⇒ 4 12 10 6 1 thus, tr4 = 6 tr10 for G =Sp(2).

3.4.4 Yang-Mills on R SU(3)/U(1) U(1) × × For the most general G-invariant ansatz (3.88) on R SU(3)/U(1) U(1) the func- × × tions λi and µi, i = 1, ..., 6, have to satisfy the G-invariance condition (3.104). One possible solution to the G-invariant ansatz (3.88) is

λ1 = λ2, λ3 = λ4, λ5 = λ6, (3.145)

µ1 = µ2, µ3 = µ4, µ5 = µ6. (3.146) Hence we can deﬁne three complex functions

φ1 = λ1 + iµ1, φ2 = λ2 + iµ2, φ3 = λ3 + iµ3, (3.147) and consequently with (see [82])

J = 1, J = 1, J56 = 1 (3.148) 12 − 34 − obtain the following expressions for Xa (compare with section 3.4.2)

X = Re(φ )I + Im(φ )I2, X2 = Im(φ )I Re(φ )I2, (3.149) 1 1 1 1 1 1 − 1 X3 = Re(φ2)I3 + Im(φ2)I , X = Im(φ2)I3 + Re(φ2)I , (3.150) 4 4 − 4 X5 = Re(φ3)I5 Im(φ3)I6, X6 = Im(φ3)I5 + Re(φ3)I6. (3.151) − Now we have to choose a matrix representation for the generators IA of the group SU(3). An appropriate basis is the 3-dimensional fundamental representation i given by the Gell-Mann-matrices, which we rescale with the factor − : √3      0 1 0   0 i 0  i   i  −  I1 = −  1 0 0  , I2 = −  i 0 0  , (3.152) √   √   2 3  0 0 0  2 3  0 0 0       0 0 1   0 0 i  i   i  −  I3 = −  0 0 0  , I4 = −  0 0 0  , (3.153) √   √   2 3  1 0 0  2 3  i 0 0       0 0 0   0 0 0  i   i   I5 = −  0 0 1  , I6 = −  0 0 i  , (3.154) √   √  −  2 3  0 1 0  2 3  0 i 0       1 0 0   1 0 0  i   i   I7 = −  0 1 0  , I8 = −  0 1 0  . (3.155) √  −  6   2 3  0 0 0   0 0 2  − 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 89 2 ×

For the matrices Xa this means      0 φ1 0   0 iφ1 0  i  ¯  i  ¯ −  X1 = −  φ1 0 0  , X2 = −  iφ1 0 0  , (3.156) √   √   2 3  0 0 0  2 3  0 0 0   ¯   ¯   0 0 φ2   0 0 iφ2  i   i  −  X   X   3 = −  0 0 0  , 4 = −  0 0 0  , (3.157) 2 √3   2 √3   φ2 0 0 iφ2 0 0      0 0 0   0 0 0  i   i   X   X  i  5 = −  0 0 φ3  , 6 = −  0 0 φ3  . (3.158) 2 √3   2 √3  −  0 φ¯ 3 0 0 iφ¯ 3 0

For the structure constants we use those given in [82] with a rescaling fABC 1 7−→ fABC: √3 1 1 f127 = , f136 = f154 = f235 = f246 = = f347 = f657, (3.159) √3 2 √3 1 f = f = . (3.160) 568 348 2

Therefore, in order to calculate the equation of motion (3.102) for X1 we take advantage of the fact that the most of structure constants are zero, especially this means that

1 1 1 fcd[Xc, Xd] = 2( f36[X3, X6] + f54[X5, X4]) 1 = ([X3, X6] + [X5, X4]). (3.161) √3 Evaluating this we get  ¯ ¯   0 φ2φ3 0  1 i   f [Xc, Xd] = −  φ2φ3 0 0  . (3.162) cd √   6 3  0 0 0 

Further, we need the terms of the form [Xa, [Xa, X1]]:

 2   0 φ1 φ1 0  i  2 | |  [X2, [X2, X1]] =  φ1 φ1 0 0  , (3.163) √  | |  6 3  0 0 0   2   0 φ1 φ2 0  i  ¯ 2 | |  [X3, [X3, X1]] = [X4, [X4, X1]] =  φ1 φ2 0 0  , (3.164) √  | |  12 3  0 0 0   2   0 φ1 φ3 0  i  ¯ 2 | |  [X5, [X5, X1]] = [X6, [X6, X1]] =  φ1 φ3 0 0  . (3.165) √  | |  12 3  0 0 0  90 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

If we now plug above results in our e.o.m. (3.102) and consider only the (1, 2) component, we get

2 2 2 6φ¨ = (κ 1)φ (κ + 3)φ¯ 2φ¯3 + φ (2 φ + φ2 + φ3 ). (3.166) 1 − 1 − 1 | 1| | | | |

Because of the symmetry in φ1, φ2 and φ3 the e.o.m. for φ2 and φ3 can be obtained via cyclic permutation of indices. It remains to derive an equation corresponding to the constraints (3.103). Using expressions for Xa, we easily obtain

   Y1 Y2 0 0  1  −  X X˙  Y Y  0 = [ a, a] = −  0 3 1 0  , (3.167) 12  −  0 0 Y2 Y3 −

˙ where Yi = φiφ¯ i φ˙ iφ¯ i. Therefore, we have −

Y Y2 = Y3 Y = Y2 Y3 = 0. (3.168) 1 − − 1 −

3.4.5 Yang-Mills on R Sp(2)/Sp(1) U(1) × × Analogously to the case on R SU(3)/U(1) U(1) we proceed the calculations × × for R Sp(2)/Sp(1) U(1). Here the G-invariance condition 3.113for the almost × × complex structure given in (3.148) has a solution of the form (see section 3.4.2)

X = Re(φ )I Im(φ )I2, X2 = Im(φ )I + Re(φ )I2, (3.169) 1 1 1 − 1 1 1 1 X3 = Re(φ )I3 Im(φ )I , X = Im(φ )I3 + Re(φ )I , (3.170) 1 − 1 4 4 1 1 4 X5 = Re(φ2)I5 Im(φ2)I6, X6 = Im(φ2)I5 + Re(φ2)I6, (3.171) − with complex valued functions φ1, φ2. As a matrix representation of the group Sp(2) we choose the four-dimensional fundamental representation given by

   0 0 0 1   −    i i  0 0 1 0  I1 = − σ2 σ2 = −   , (3.172) 2 √6 × 2 √6  0 1 0 0     1 0 0 0   −   0 0 0 i   −    i i  0 0 i 0  I2 = − σ2 σ1 = −  −  , (3.173) 2 √6 × 2 √6  0 1 0 0     1 0 0 0  3.4. YANG-MILLS ON G -MANIFOLDS R G/H 91 2 ×

   0 0 i 0   −    i i  0 0 0 i  I3 = − σ2 σ3 = −   , (3.174) 2 √6 × 2 √6  i 0 0 0     0 i 0 0   −   0 0 1 0   −    i i  0 0 0 1  I4 = − σ1 Id2 = −  −  , (3.175) 2 √6 × 2 √6  1 0 0 0   −   0 1 0 0  −    0 0 0 0  !     i 0 0 i  0 0 0 0  I5 = − σ2 = −   , (3.176) 2 √3 0 1 × 2 √3  0 0 0 i   −   0 0 i 0 

   0 0 0 0  !     i 0 0 i  0 0 0 0  I6 = − σ1 = −   , (3.177) 2 √3 0 1 × 2 √3  0 0 0 1     0 0 1 0 

   0 0 0 0  !     i 0 0 i  0 0 0 0  I7 = − σ3 = −   , (3.178) 2 √3 0 1 × 2 √3  0 0 1 0     0 0 0 1   −   0 i 0 0  !  −    i 1 0 i  i 0 0 0  I8 = − σ2 = −   , (3.179) 2 √3 0 0 × 2 √3  0 0 0 0     0 0 0 0 

   0 1 0 0  !     i 1 0 i  1 0 0 0  I9 = − σ1 = −   , (3.180) 2 √3 0 0 × 2 √3  0 0 0 0     0 0 0 0     1 0 0 0  !     i 1 0 i  0 1 0 0  I10 = − σ3 = −  −  , (3.181) 2 √3 0 0 × 2 √3  0 0 0 0     0 0 0 0  In this way we obtain     0 0 0 φ¯ 0 0 0 iφ¯  1   1   −   −  i  0 0 φ1 0  i  0 0 iφ1 0  X1 = −  , X2 = −  − , (3.182) √  0 φ¯ 1 0 0  √  0 iφ¯ 1 0 0  2 6   2 6    φ 0 0 0   iφ 0 0 0  − 1 1 92 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

    0 0 iφ 0 0 0 φ 0  1   1   − ¯   − ¯  i  0 0 0 iφ1  i  0 0 0 φ1  X3 = −  , X4 = −  − , (3.183) √  iφ¯ 1 0 0 0  √  φ¯ 1 0 0 0  2 6   2 6  −   0 iφ 0 0   0 φ 0 0  − 1 − 1      0 0 0 0   0 0 0 0      i  0 0 0 0  i  0 0 0 0  X   X   5 = −  , 6 = −   (3.184) 2 √3  0 0 0 iφ2  2 √3  0 0 0 φ2   −    0 0 iφ¯ 2 0 0 0 φ¯ 2 0

For the structure constants we use those given in [82] rescaling f 1 f . So −→ √3 that the only non-vanishing components are f 5 = f 5 = f 6 = f 6 = 1 . 41 32 13 42 2 √3 In order to evaluate the Yang-Mills equations (3.102) and (3.103), we remark that there is no symmetry in φ1 and φ2, so that consequently the equations of motions and constrains for Xa, a = 1, ..., 4 and Xb, b = 5, 6, should be considered separately. But at ﬁrst we calculate the relevant terms:

1 1 1 fcd[Xc, Xd] = 2( f36[X3, X6] + f54[X5, X4]) (3.185) 1 = ([X3, X6] + [X5, X4]) (3.186) √3    0 0 0 φ1φ2   ¯ ¯ −  i  0 0 φ1φ2 0  = −   . (3.187)  ¯ ¯  6 √6  0 φ1φ2 0 0    φ φ2 0 0 0 − 1

Further, we need cubic terms of type [Xa, [Xa, X1]]:

 ¯ 2   0 0 0 φ1 φ1   − | |   2  i  0 0 φ1 φ1 0  [X2, [X2, X1]] =  2 | |  , (3.188) 4 √6  0 φ1 φ1 0 0   | |   φ¯ φ 2 0 0 0  − 1| 1|

[X3, [X3, X1]] = [X4, [X4, X1]] = 2[X5, [X5, X1]] = 2[X6, [X6, X1]] (3.189)  ¯ 2   0 0 0 φ1 φ2   2 − | |  i  0 0 φ1 φ2 0  =  | |  .(3.190)  2  36 √6  0 φ1 φ2 0 0   2 | |  φ¯ φ2 0 0 0 − 1| | Plugging all the terms together in (3.102) and focusing on the (2, 3) component, we arrive at

2 2 6φ¨ = (κ 1)φ (κ + 3)φ¯ φ¯ 2 + φ (3 φ + φ2 ). (3.191) 1 − 1 − 1 1 | 1| | | 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 93 2 ×

Calculation for X6 gives then with

6 6 6 fcd[Xc, Xd] = f13[X1, X3] + f42[X4, X2] (3.192)    0 0 0 0    i  0 0 0 0  = −   , (3.193)  0 0 0 φ¯ 2  6 √3  1   2  0 0 φ1 0

[X1, [X1, X6]] = [X2, [X2, X6]] = [X3, [X3, X6]] = (3.194)   0 φ¯ 2φ¯ 0 0  1 2   2  i  φ φ2 0 0 0  = [X , [X , X ]] =  1  , (3.195) 4 4 6  2  6 √3  0 0 0 φ2 φ1   2 | |  0 0 φ¯ 2 φ1 0  | |  0 φ¯ φ¯ 2 0 0  2 1   2 −  i  φ φ2 0 0 0  [X , [X , X ]] =  − 1  , (3.196) 2 2 6  2  6 √3  0 0 0 φ2 φ1   2 | |  0 0 φ¯ 2 φ1 0  | |  0 φ2φ 0 0  1 2   ¯ 2 ¯  i  φ φ2 0 0 0  [X , [X , X ]] =  1  , (3.197) 3 3 6  2  6 √3  0 0 0 φ2 φ1   2 | |  0 0 φ¯ 2 φ1 0  | |  0 φ2φ 0 0  1 2   ¯ 2 ¯ −  i  φ φ2 0 0 0  [X , [X , X ]] =  − 1  , (3.198) 4 4 6  2  6 √3  0 0 0 φ2 φ1   2 | |  0 0 φ¯ 2 φ1 0  | |  0 0 0 0    i  0 0 0 0  [X , [X , X ]] =   , (3.199) 5 5 6  2  6 √3  0 0 0 φ2 φ2   2 | |  0 0 φ¯ 2 φ2 0 | | the following result

κ 1 κ + 3 2 1 2 2 φ¨ 2 = − φ2 φ¯ + φ2( φ + φ2 ). (3.200) 6 − 6 1 3 | 1| | | Constraints

[Xa, X˙ a] = 0, (3.201) also easily obtained with    Y1 0 0 0      ˙ ˙ 1  0 Y1 0 0  [X1, X1] = [X2, X2] = −  −  , (3.202) 24  0 0 Y1 0     0 0 0 Y  − 1 94 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

   Y1 0 0 0   −    ˙ ˙ 1  0 Y1 0 0  [X3, X3] = [X4, X4] = −   , (3.203) 24  0 0 Y1 0     0 0 0 Y  − 1

   0 0 0 0    1  0 0 0 0  [X , X˙ ] = [X , X˙ ] = −   (3.204) 5 5 6 6  Y  12  0 0 2 0   −  0 0 0 Y2 with

˙ Yi = φ˙ iφ¯ i φiφ¯ i. (3.205) − Therefore, from the constraint (3.201) can be written as

˙ ˙ Y Y2 = 0 or φ˙ φ¯ φ φ¯ = φ˙ 2φ¯ 2 φ2φ¯ 2. (3.206) 1 − 1 1 − 1 1 − Remark: We observe that the equation of motion (3.191) for the case R × Sp(2)/Sp(1) U(1) can be derived from e.o.m. computed for R SU(3)/U(1) U(1) × × × via identiﬁcation φi φ2, φj φ and φ φ for i, j, k 1, 2, 3 , i , j , k. −→ −→ 1 k −→ 1 ∈ { }

3.4.6 Action

In the same way as it was presented in [2], we will now see that the Yang- Mills equations with torsion (3.263) and (3.191) can be derived from an action functional, concretely of the form Z 1 S = tr + tr(κe0 ω ). (3.207) F ∧ ∗F 3 ∧ ∧ F ∧ F R G/H ×

First of all our goal is to evaluate this action in terms of Xa. For that reason rewrite it as follows, κ S = S + S , (3.208) 1 3 2 with Z S = tr( ), (3.209) 1 F ∧ ∗F R G/H ×Z 0 S2 = tr(e ω ). (3.210) ∧ ∧ F ∧ F R G/H × 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 95 2 ×

Then we have Z 1 ab 0 1 2 3 4 5 6 S = tr(F F + 2F0aF0a)e e e e e e e (3.211) 1 2 ab ∧ ∧ ∧ ∧ ∧ ∧ R G/H × Z 1 = Vol(G/H) dτtr(F Fab + 2F F0a). (3.212) 2 ab 0a R Recall that according to (3.89)

i c F = ( f Ii + f Xc [Xa, X ]), F0a = X˙ a and (3.213) ab − ab ab − b i A0 = 0, Aa = eaIi + Xa. (3.214)

Then we have Z 1 j S = Vol(G/H) dτtr(2X˙ X˙ + f i f I I + f c f d X X + 1 2 a a ab ab i j ab ab c d R i c i +[Xa, X ][Xa, X ] 2 f f IiXc + 2 f Ii[Xa, X ] + b b − ab ab ab b c +2 fabXc[Xa, Xb]), (3.215) where we used the property tr(AB) = tr(BA). Then, since

c d c d 1 fab fab = fib fib = 3 δcd, (3.216) c tr(Ii[Xa, X ]) = tr([Ii, X ]Xa) = f tr(XcXa), (3.217) b − b − ib and

tr(IiXa) = (λaδab + µa Jab)tr(IiIb) = 0, (3.218) because of Ii h m we get ∈ ⊥ Z 1 i j 1 S = Vol(G/H) dτtr(2X˙ aX˙ a + f f IiIj XcXc 1 2 ab ab − 3 − R c 2 f Xc[Xa, X ] + [Xa, X ][Xa, X ]) (3.219) − ab b b b 1 a b In order to calculate S2 we recall that ω ω ω = 6vol6 and ω = J e e . Thus, ∧ ∧ 2 ab ∧

1 a b c d e f 6vol6 = J J J e e e e e e = (3.220) 8 ab cd e f ∧ ∧ ∧ ∧ ∧ 1 = J J J abcde f vol . (3.221) 8 ab cd e f 6 abcde f 1 abcde f The relevant term appearing in the integral S2 is ωab = 2 Jab . So we extract it from (3.221) taking in mind that it is total antisymmetric in indices. We ﬁnd 1 J abcde f = J(cd Je f ), (3.222) 2 ab 96 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS where denote by J(cd Je f ) the totally antisymmetric combination Jcd Je f Jce Jd f − − Jc f Jed. A proof conﬁrms (compare with (3.221)) 1 1 J J J J vol6 = J J (J J Jce J J J ) = (3.223) 4 cd e f (cd e f ) 4 cd e f cd e f − d f − c f ed 1 = (δcδe δc δc δe δe ) = (3.224) 4 c e − f f − d d 1 = (36 6 6) = 6vol6. (3.225) 4 − −

To evaluate the under integral expression of S2 we substitute (3.222) and (3.89) into it:

κ abcde f tr(dτ ω ) = tr( J F F )vol7 = (3.226) ∧ ∧ F ∧ F 8 ab cd e f 1 = tr(J J F F )vol . (3.227) 4 (cd e f ) cd e f 7 i i i ˜ Because of Jab fab = 0, Jab fcb = Jcb fab and the antisymmetry of fabc = fabd Jdc and of J we obtain

Jcd Je f FcdFe f = 0 and Jce Jd f FcdFe f = Jc f JedFcdFe f . (3.228)

Therefore,

1 d f i c i a tr(dτ ω ) = tr(Jce J ( f Ii + f Xc [Xa, X ])( f Ii + f Xa ∧ ∧ F ∧ F −2 ab ab − b cd cd − k b [Xc, X ])( f I + f X [Xe, X ]))vol7 = (3.229) − d e f k e f b − f 1 i k i = Jce J ( f f tr(IiI ) 2 f tr(Ii[Xe, X ]) −2 d f cd e f k − cd f − b a b 2 f tr(X [Xe, X ]) + f f tr(XaX ) + − cd b f cd e f b +tr([Xc, Xd][Xe, X f ]))vol7. (3.230)

At this point we calculate

i k i k i i Jce Jd f fcd fe f = fcd fcd, Jce Jd f fcd = fe f , (3.231)

b b a b 1 ab Jce J f = f , Jce J f f = δ , (3.232) d f cd − e f d f cd e f −3 J J tr([Xc, X ][Xe, X ]) = J J tr(Xc[[Xe, X ], X ]) = (3.233) (cd e f ) d f − (cd e f ) f d = J J (tr(Xc[[Xe, X ], X ] cd e f f d − Xc[[X , X ], Xe] Xc[[X , Xe], X ]) =(3.234) − d f − f d = Jcd Je f (tr(Xc[[Xe, X f ], Xd] +

+Xc[[X f , Xd], Xe] + Xc[[Xd, Xe], X f ]) =(3.235) = 0, (3.236) i i i b f tr(Ii[Xe, X ]) = f tr([Ii, X ]Xe) = f f tr(X Xe) = (3.237) e f f − e f f − e f i f b 1 1 = δ tr(X X ) = tr(X X ). (3.238) 3 eb b e 3 b b 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 97 2 ×

Hence,

1 i k tr(dτ ω ) = ( f f tr(IiI ) + 2 f tr(Xa[X , Xc]) ∧ ∧ F ∧ F −2 cd cd k abc b − tr(XaX ))vol7. (3.239) − b

Plugging (3.239) in the expression for S2, we get Z 1 i k S2 = Vol(G/H) dτtr( f f IiI 2 f Xa[Xe, X ] + XaXa). (3.240) 2 − cd cd k − ae f f R

So that for the action functional it follows Z κ 1 i j S = S + S2 = Vol(G/H) dτtr(X˙ aX˙ a + f f IiIj 1 3 2 ab ab − R

1 c 1 XcXc f Xc[Xa, X ] + [Xa, X ][Xa, X ]) + −6 − ab b 2 b b Z κ i k + Vol(G/H) dτtr(X X 2 f Xa[X , Xc] f f IiI ) (3.241) 6 b b − abc b − cd cd k R Z κ 1 κ + 3 c = Vol(G/H) dτtr(X˙ aX˙ a + − X X f Xc[Xa, X ] + 6 b b − 3 ab b R 1 3 κ j + [X , X ][X , X ]) + − f i f I I ). (3.242) 2 a b a b 6 ab ab i j Varying this action, one can see that the Euler-Lagrange equations for it are (3.102). Remark: The action, in particular the integral S2, can also be evaluated by using the Chern-Simon form of (see for example [14,5]). F ∧ F

3.4.7 Action for the case R SU(3)/U(1) U(1) × × To get the action in terms of φ1, φ2 and φ3, let us substitute the exact form of Xa for the case SU(3)/U(1) U(1) the into (3.242). We take the matrices given in × (3.156-3.158). Then we obtain

1 2 2 2 tr(XaXa) = ( φ + φ2 + φ3 ), (3.243) −3 | 1| | | | | a 1 f (Xa[X , Xc]) = (φ φ2φ3 + φ¯ φ¯ 2φ¯ 3), (3.244) bc b −6 1 1 1 4 4 4 tr([Xa, X ][Xa, X ]) = ( φ + φ2 + φ3 + b b −9 | 1| | | | | 2 2 2 2 2 2 + φ φ2 + φ φ3 + φ2 φ3 ), (3.245) | 1| | | | 1| | | | | | | 1 j f i f tr(I I ) = a2. (3.246) 2 ab ab i j 98 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

The constant term a2 can be evaluated according to the previous section (3.4.3), where we ﬁnd that in case of the gauge group SU(3) the trace over IAIB satisﬁes 1 trIAIB = χ0 δAB = δAB, (3.247) − R −6 so that

2 1 i j 1 i j ij 1 i i 1 1 aa 1 a = f f tr(IiIj) = f f δ = f f = δ = . (3.248) 2 ab ab −12 ab ab −12 ab ab −12 3 −6 Substituting (3.243)-(3.247) into (3.242) we obtain Z X X 1 2 κ 1 2 S = Vol(G/H) dτ φ˙i − φi + − 3 | | − 18 | | R i i X κ + 3 1 4 + (φ φ2φ3 + φ¯ φ¯ 2φ¯ 3) φi 18 1 1 − 18 | | − i 1 2 2 2 2 2 2 κ 3 ( φ φ2 + φ φ3 + φ2 φ3 + − . (3.249) −18 | 1| | | | 1| | | | | | | 6 Writing this action in terms of a potential V(φ),   Z X  1 2 1  S = Vol(G/H) dτtr  φ˙ V(φ) , (3.250) −3 | | − 2  i R with X X κ 1 2 κ + 3 1 4 V(φ) = − φi (φ φ2φ3 + φ¯ φ¯ 2φ¯ 3) + φi + 9 | | − 9 1 1 9 | | i i 1 2 2 2 2 2 2 κ 3 + ( φ φ2 + φ φ3 + φ2 φ3 ) − , (3.251) 9 | 1| | | | 1| | | | | | | − 3 one ﬁnds that the e.o.m (3.263) then reads ∂V 6φ¨i = 9 . (3.252) ∂φ¯ i

3.4.8 Action for the case R Sp(2)/Sp(1) U(1) × × Again using the explicit form of the matrices Xa, we compute the action but now for the case R Sp(2)/Sp(1) U(1). Plugging the corresponding matrices × × (3.182-3.184) into the action integral (3.242), we obtain that

1 2 2 tr(XaXa) = (2 φ + φ2 ), (3.253) −3 | 1| | | a 1 2 2 f (Xa[X , Xc]) = (φ¯ φ¯ 2 + φ φ2), (3.254) bc b −6 1 1 1 4 4 2 2 tr([Xa, X ][Xa, X ]) = (3 φ + φ2 + 2 φ φ2 ), (3.255) b b −9 | 1| | | | 1| | | 1 i j 2 f f tr(I I ) = a0 , (3.256) 2 ab ab i j 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 99 2 × with

2 1 i j 1 i j 1 1 a0 = f f χ0 δij = f f δij = . (3.257) −2 ab ab R −2 ab ab 6 −6 Consequently we arrive at Z 1 2 2 S = Vol(G/H) dτ( (2 φ˙ + φ˙ 2 ) −3 | 1| | | − R κ 1 2 2 κ + 3 ¯ 2 ¯ 2 − (2 φ + φ2 ) + (φ1φ2 + φ1φ2) − 18 | | | | 18 − 1 4 4 2 2 κ 3 (3 φ + φ2 + 2 φ φ2 ) + − . (3.258) −18 | 1| | | | 1| | | 6 Decomposing again the integrand in kinetic and potential terms Z 1 2 2 1 S = Vol(G/H) dτ( (2 φ˙ + φ˙ 2 ) V(φ1, φ2)), (3.259) −3 | | | | − 2 R with κ 1 2 2 κ + 3 ¯ 2 ¯ 2 V(φ) = − (2 φ + φ2 ) (φ1φ2 + φ1φ2) 9 | | | | − 9 − 1 4 4 2 2 κ 3 (3 φ + φ2 + 2 φ φ2 ) − , (3.260) −9 | 1| | | | 1| | | − 3 one gets ∂V ∂V 4φ¨ 1 = 3 , 6φ¨ 2 = 9 . (3.261) ∂φ¯ 1 ∂φ¯ 2

3.4.9 Yang-Mills on R G /SU(3) × 2 The most general G-invariant ansatz (3.88) for the connection one-form on A R G2/SU(3) has the form (see 3.104) × i a = e Ii + e X (τ)I with X = φ Id + φ2 J . (3.262) A ab b ab 1 ab ab We identify X with the complex valued function φ = φ1 + iφ2. This case was par- ticularised in [2], but can also be derived directly from the already discussed case R SU(3)/U(1) U(1). For this purpose we should only put equal the functions × × φ := φ1 = φ2 = φ3. In this way we obtain from (3.263) the Yang-Mills equation for R G2/SU(3), namely: × 6φ¨ = (κ 1)φ (κ + 3)φ¯ 2 + 4φ φ 2. (3.263) − − | | The constrains (3.168) then are fulﬁlled identically. For the corresponding potential we obtain κ 1 κ + 3 2 κ 3 V(φ) = − φ 2 (φ3 + φ¯ 3) + φ 4 − (3.264) 3 | | − 9 3| | − 3 and hence, ∂V 6φ¨ = 3 . (3.265) ∂φ¯ 100 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

3.4.10 Yang-Mills on R S3 S3 × × In this work we will not discuss the Yang-Mills equations on R S3 S3, the last × × relevant case for possible nearly Kahler¨ structure, but instead refer the reader to [83]. There one can ﬁnd the complete discussion this case.

3.4.11 Solutions on R SU(3)/U(1) U(1) × × The following sections we dedicate to solutions of the Yang-Mills equations (3.263), which can be carried out thanks to a possibility in some certain cases to introduce a “superpotential”, due to which the second order differential equations deduce from first order differential equation. The latter, which we call the gradient or Hamiltonian flow equations, can be solved much easier.

Critical points of the potential

Let us consider the equation of motion found for the case R SU(3)/U(1) U(1): × ×

2 2 2 ∂V 6φ¨ 1 = (κ 1)φ1 (κ + 3)φ¯ 2φ¯3 + φ1(2 φ1 + φ2 + φ3 ) = 9 . (3.266) − − | | | | | | ∂φ¯ 1

We observe that this equations describe the motion of a particle moving in a three-dimensional complex space under the action of the potential V. Note − that only for real κ the potential is real and this mechanical interpretation is just straightforward. First of all we are interested in instanton solutions corresponding to a particle moving form critical points of V to critical points. Due to conservation of energy, such a trajectory can exist only if V(τ ) = V(τ ). Searching for the −→ ∞ −→ −∞ extrema φˆ of the potential V we find the following points: where ν = 1 (κ 1). The continuous symmetry (let us call it the 3α-symmetry) 4 − q ˆ ˆ ˆ 1 κ iα iα1 iα2 iα3 (φ1, φ2, φ3) (0, 0, 0) (0, 0, −2 e )(e , e , e )(ν, ν, ν) 2 (κ+1+2 √3)(κ+1 2 √3) 2 3 V (1 2ν) − 0 (1 + ν)(1 ν) 9 − − 36 9 − iα P φi φie , where αi = 0 mod 2π, as well as the permutation symme- −→ i try applied on the given (φˆ , φˆ , φˆ ) produce further extreme points. Remark q 1 2 3 1 κ iα that (0, 0, −2 e ) is a local maximum for κ < 1 with respect to φ3, since iα p ∂ ∂ ¯ V(0, 0, e (1 κ)/2) < 0 for κ < 1. φ3 φ3 − Following the notation in [2], we differ between transverse trajectories, connecting one of the critical point with that critical point descending from the first one by the 3-symmetry φ exp( 2 πi), and the radial trajectories, connecting 7−→ 3 two of the critical points with φˆ on the real axis degenerated in energy. As we have found there, similar to the case R G2/SU(3) discussed in [2], the radial × 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 101 2 × trajectories are possible for κ = 3, 1 2 √3, 1, 3, 9. The transverse trajectories − − ± occur for κ = 1, 7 (i.e., for ν = 1 , 2). − − − 2 − The action of an instanton configurations is finite only if V(τ = ) = 0. ±∞ Comparing with the results of [2], this is the case for κ = 3, 1 2 √3, 3 (i.e., for − − − ν = 1, 1 , 1 ). − − 2 2 Physical meaning of the constraints

Let us take a look at the constraints

Yi Yj = 0, i, (3.267) − ∀ ˙ where Yi = φiφ¯ i φ˙ iφ¯ i. First of all we observe that due to (3.251) − ¨ Y˙ i = φiφ¯ i φ¨ iφ¯ i = (3.268) − ! 3 ∂V ∂V κ + 3 = φi φ¯ i = (φ¯ 1φ¯ 2φ¯ 3 φ1φ2φ3) (3.269) 2 ∂φi − ∂φ¯ i 2 −

Y˙ i is index independent. It follows that Y˙ i Y˙ j = 0 and Yi Yj is always conserved. −1 − P Consequently, also conserved is Y˜ i = Yi (Y +Y2 +Y3) with Y˜i = 0 or Yi = Yj. − 3 1 i Suppose that Y1 +d = Y2, Y2 +d˜ = Y3. However, the constraints (3.267) mean that there are no additional constants d, d˜. Therefore, they ﬁx the symmetry between the φi, i = 1, 2, 3.

Duality

The surprising duality appearing in the case R G2/SU(3) and discussed in [2] × can also be observed on the solutions of the equations of motion (3.263). In order to see that we write the equations in terms of ν:

2 2 2 6φ¨ i = 4νφi 4(ν + 1)φ¯ jφ¯ + φi(2 φi + φj + φ ), (3.270) − k | | | | | k| X 2 4 2 8 3V(φ1, φ2, φ3) = (1 2ν) + φn (1 + ν)Reφ1φ2φ3 + 3 − 3 n | | − 3 X 2 4 + φn , (3.271) 3 n | | with i , j , k. The duality consists in the correspondence between solutions of (3.270) connected via the map

1 1 τ 1 τ 1 τ (ν, φ (τ), φ2(τ), φ3(τ)) , φ , φ2 , φ3 . (3.272) 1 7−→ ν ν 1 ν ν ν ν ν

Gradient ﬂow

We consider the case κ = 3. The duality transformation makes it possible to apply the result also to κ = 9. Moreover, accomplishments obtained in [2] can be 102 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

used here. Concretely, we also find out that the special case φ1 = φ2 = φ3 of the e.o.m. (3.263) are implied by the first order gradient flow equations expressed by a superpotential W (they also resolve the constrains (3.267)):

√ ˙ ¯ 2 ∂W 3φi = φi φi = , (3.273) ± − ∂φ¯ i φ1 = φ2 = φ3, (3.274) where X 2 W = φ φ2φ3 + φ¯ φ¯ 2φ¯ 3 φ . (3.275) 1 1 − | i | i

Note that W and the flow equation (3.273) are invariant under 3α-symmetry. The finite-action kink solutions then are ! 1 τ τ0 τ τ0 τ τ0 (φ1, φ2, φ3) = 1 tanh − , 1 tanh − , 1 tanh − , (3.276) 2 ± 2 √3 ± 2 √3 ± 2 √3 with the collective coordinate τ0. Note that the 3α-symmetry gives further solutions. The explicit infinite-action solutions of the gradient flow were already given in [2]. So we only briefly give the final outcome. For the polar decomposition φ (τ) = r(τ)exp(iϕ(τ)) and φ = φ2 = φ with the corresponding 1 | 1| | | | 1| 3α symmetry the solution is −

1 1/3 1 5 3 2 r(ϕ) = (sin 3ϕ)− [cos 3ϕ 2F ( , , , cos 3ϕ) + C], (3.277) −3 1 2 6 2 for some real constant C and the hypergeometric functions 2F1. This trajectories are unbounded.

Continuous symmetry

We consider the case κ = 3. It is special due to the continuous symmetry arising − from the fact that κ = 3 is ﬁxing point of the duality transformation and from − the U(1)3 symmetry of the solutions of the e.o.m. (3.263). As before, we found a superpotential W only for solutions of a special type, φ = φ2 = φ3 . Namely, | 1| | | | | X W = 2 φ φ2 φ3 2 φi . (3.278) | 1|| || | − | | i

With that W the second order equations of motion for φi follow from the gradient ﬂow equations

φi ∂W iα iβ √3φ˙ i = ( φj φk 1) = , φ2 = e φ1, φ3 = e φ1, (3.279) ± φi | || | − ∂φ¯ | | i 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 103 2 × with arbitrary α, β [0, 2π]. Finite-action solutions we extract from [2] are ∈ ! iα τ τ0 iβ τ τ0 iγ τ τ0 (φ1, φ2, φ3) = e tanh − , e tanh − , e tanh − , (3.280) √3 √3 √3 where α, β, γ are arbitrary angles. The infinite-action solution have also been given. The corresponding trajectories connect the critical S1 S1 S1 torus with × × the point at the infinity ( , , ): ∞ ∞ ∞ iα τ τ0 iβ τ τ0 iγ τ τ0 (φ1, φ2, φ3) = (e coth − , e coth − , e coth − ), (3.281) ∓ √3 √3 √3 with arbitrary α, β, γ. There are only these radial solutions of the gradient flow equations. Now we turn to an other case. We consider solutions of (φ1, φ2, φ3), where some two φi vanish identically. Suppose, it is φ1 which remains in the differential equations. So the equation of motion takes the form

2 6φ¨ = (κ 1)φ + 2φ φ , φ2 = φ3 0. (3.282) 1 − 1 1| 1| ≡

In particular, there is no coupling between φi in the differential equation. Com- paring with the previous solution one finds that the superpotential for this special case is X X √2 3 1 W = φi + (κ 1) φi , (3.283) 3 | | − | | i √2 i then the e.o.m. (3.282) follows from the gradient flow equation 1 φ1 2 κ 1 ∂W √3φ˙ 1 = φ1 + − = . (3.284) ± √ φ | | 2 ∂φ¯ 2 | 1| 1 q 1 κ ˜ Supposing φ1 = −2 φ1, we again come back to (3.279) and obtain as a radial trajectory solution r ! iα 1 κ φ˜ = e tanh − (τ τ0) , α [0, 2π], (3.285) 1 12 − ∈ or r r ! iα 1 κ 1 κ φ = e − tanh − (τ τ0) , α [0, 2π], φ2 = φ3 0. (3.286) 1 2 12 − ∈ ≡ q iα 1 κ For κ > 1 the trajectory is periodic and non-continuous. Since V(e −2 , 0, 0) = (κ+1+2 √3)(κ+1 2 √3) − the solutions (3.286) have finite action only if − 36 κ = 1 2 √3. (3.287) − − 104 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

Another solution of the gradient flow equation (3.284) is r r ! 1 κ iα 1 κ φ = − e coth − (τ τ0) , α [0, 2π], φ2 = φ3 0. (3.288) 1 ∓ 2 12 − ∈ ≡ For κ < 1 it corresponds to a infinite-action solution and to a trajectory connecting the critical circle (eiα, 0, 0) with the point at the infinity ( , 0, 0). For κ > 1 this ∞ solution yields a periodic infinite-action solution if κ , 1 + 2 √3, otherwise the − action is finite and the trajectory is periodic. At last we consider the case κ = 1. Then equation of motion takes the form

6φ¨ = 2φ φ , φ2 = φ3 0. (3.289) 1 1| 1| ≡ Using the same superpotential (3.283), we obtain the gradient ﬂow equation

1 ∂W √3φ˙ = φ φ = . (3.290) 1 1 1 ¯ ± √2 | | ∂φ1

iϕ(τ) In polar representation φ1 = r(τ)e ) we then have

√6(r˙ + irϕ˙ ) = r2. (3.291) ± Hence, √6r˙ = r2, ϕ˙ = 0 and ϕ is constant. We find a solution ± √6 r(τ) = , τ0 > 0. (3.292) τ + τ0 | | We remark that this solution is not differentiable at the origin. The trajectory √ of this radial infinite-action solution φ (τ) = 6 eiϕ, φ = φ = 0 connects the 1 τ +τ0 2 3 origin with itself. | | Note that for all cases mentioned above further solutions are obtained via 3α-symmetry.

Hamiltonian ﬂow

Now let us consider transverse trajectories. For that goal we again use solutions presented in [2]. Concretely, for φi connected by the 3α-symmetry and φ = | 1| φ2 = φ3 and for κ = 1 (κ = 7 yields a solution, which can be obtained by the | | | | − duality transformation) there is a superpotential X 2 W = φ φ2φ3 + φ¯ φ¯ 2φ¯ 3 φi , (3.293) 1 1 − | | i such that e.o.m. (3.263) are a conclusion of the ﬁrst-oder diﬀerential equations

2 ∂W √3φ˙ = (φ¯ φi) = i , (3.294) ± − ∂φ¯ φ φ = φ2 = φ3. (3.295) ≡ 1 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 105 2 ×

The ﬁnite-action solutions are 1 τ τ0 φ = 1 i √3 tanh − , i, (3.296) 2 ± 2 ∀ and those obtained via 3α symmetry. The trajectory describing the solution 4 πi 4 πi 4 πi 2 πi 2 πi 2 πi (3.296) connects the points (e 3 , e 3 , e 3 ) and (e 3 , e 3 , e 3 ). The inﬁnite-action solution is in implicit form can be given by the cubic equation (see [2])

1 r2 + C cos 3ϕ = , (3.297) 2 r3 where we use the polar representation φ = r exp(iϕ)(φ1 = φ2 = φ3 plus 2α- symmetric solutions) and C is an invariant preserved by the ﬂow equation (3.294), 1 r2+C 10 C = W((φ , φ2, φ3)(τ)). Periodic trajectories exists for 3 1, i.e., for < 1 | 2 r | ≤ − 27 C < 0.

3.4.12 Solutions on R Sp(2)/Sp(1) U(1) × × Critical points

The equations of motion on R Sp(2)/Sp(1) U(1) are × ×

2 2 9 ∂V 6φ¨ 1 = (κ 1)φ1 (κ + 3)φ¯ 1φ¯2 + φ1(3 φ1 + φ2 ) = , (3.298) − − | | | | 2 ∂φ¯ 1 ¨ 2 2 2 ∂V 6φ2 = (κ 1)φ2 (κ + 3)φ1 + 2φ2( φ1 + φ2 ) = 9 , (3.299) − − | | | | ∂φ¯ 2 with

κ 1 2 2 κ + 3 2 2 V(φ) = − (2 φ + φ2 ) (φ¯ φ¯ 2 + φ φ2) 9 | 1| | | − 9 1 1 1 4 4 2 2 κ 3 (3 φ + φ2 + 2 φ φ2 ) − , (3.300) −9 | 1| | | | 1| | | − 3 We remember that there is a correspondence to the e.o.m. on R SU(3)/U(1) U(1) × × via the projection φ φ , φ2 φ and φ3 φ2. So that the solution space 1 −→ 1 −→ 1 −→ of (3.298) and (3.299) is a subspaces of solutions of (3.263). Therefore we can read oﬀ the critical points (φˆ 1, φˆ 2) of the potential V from the previous section: 1 where 2α + α2 = 0 mod 2π and ν = (κ 1). The 3α-symmetry as well as 1 4 − q ˆ ˆ 1 κ iα iα1 iα2 iα1 iα2 (φ1, φ2) (0, 0) (0, −2 e )(e , e )(νe , νe ) 2 (κ+1+2 √3)(κ+1 2 √3) 2 3 V (1 2ν) − 0 (1 + ν)(1 ν) 9 − − 36 9 − permutation symmetry are now missing, but instead we have a symmetry, let call it 2α-symmetry, transforming solutions φ1, φ2 respectively to the solutions iα iα2 e 1 φ1 and e φ2, with 2α1 + α2 = 0. 106 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

Gradient and Hamiltonian ﬂows

Obviously, any solution of (3.298) and (3.299) is also a solution of (3.263), so it is a subcase that we discuss here. At first we observe that the simplest way to get some solutions for (3.298) and (3.299) is to narrow the solution set of the e.o.m. for the case R SU(3)/U(1) U(1), which we constructed sections. It is done × × by reducing the 3α-symmetry to the 2α-symmetry, because then we have the same differential equations (3.298) and (3.299). It is clear that for the relatively trivial case, when φ and φ2 are equal, the second order Yang-Mills equation | 1| | | are consequences of the same flow equations as before for three functions φi. Nevertheless, here appears a question whether the flow equations are reasoned by the corresponding superpotentials. It is easy to see that it indeed the fact, but only with a little alteration

1 ∂W ∂W √3φ1 = , √3φ2 = . (3.301) ± 2 ∂φ¯ 1 ± ∂φ¯ 2

The superpotential remains unchanged, one has only to identify φ φ , φ2 1 −→ 1 −→ φ and φ3 φ2. 1 −→ 3.4.13 Instanton-anti-instanton chains and dyons Periodic solutions

Looking for periodic solutions, we again assume that φ = φ1 = φ2 = φ3 so that the equations of motions take the simplest form

6φ¨ = (κ 1)φ (κ + 3)φ¯ 2 + 4φ φ 2. (3.302) − − | |

After writing the function φ as φ =: ϕ1 +iϕ2, we may consider trajectories of ϕ1(τ) unchanging with τ. Then we have to ﬁnd appropriate values for the constants ϕ1 and κ so that also ϕ¨1 = 0 is satisﬁed for all ϕ2. From (3.302) we obtain

6ϕ¨ = (κ 1)ϕ (κ + 3)ϕ2 + 4ϕ3 + [(κ + 3) + 4ϕ ]ϕ2 (3.303) 1 − 1 − 1 1 1 2 and get exactly three solutions

1 (ϕ , κ) = (0, 3), ( , 1) and (1, 7) . (3.304) 1 − −2 − −

Substituting these values into (3.302), we obtain the equations for ϕ2:

2 a) (ϕ1, κ) = (0, 3) : 2ϕ¨2 = (1 ϕ2)ϕ2 , 1− − 3 − 2 b) (ϕ1, κ) = ( 2 , 1) : 2ϕ¨2 = ( 4 ϕ2)ϕ2 , (3.305) − − − − 2 c) (ϕ , κ) = (1, 7) : 2ϕ¨2 = (3 ϕ )ϕ2 . 1 − − − 2 Here, we rescaled τ 2 τ. → √3 3.4. YANG-MILLS ON G -MANIFOLDS R G/H 107 2 ×

We can use solutions of this type of equations, which we already treated in section 3.4.12, in order to obtain a periodic kind of solutions (sphalerons). For that purpose we require φ to be cyclic:

ϕ1(τ + L) = ϕ1(τ) and ϕ2(τ+L) = ϕ2(τ) . (3.306) This condition is equivalent to the consideration of function φ on a circle S1 with circumference L. Sphaleron solutions to three equations (3.305) can be described simultaneously by √3 ϕ = γ 2k b(k) sn[b(k)γτ; k] with γ = 1, , √3 , (3.307) 2 2 respectively, where b(k) = (2 + 2k2) 1/2 and 0 k 1. The Jacobi elliptic function − ≤ ≤ sn[u; k] has a period of 4K(k) (see appendix A.2), the condition (3.306) is satisfied for b(k) L = 4K(k) n for n N , (3.308) ∈ (n) which fixes k = k(L, n) so that ϕ2(τ; k) =: ϕ (τ). Solutions (3.307) exist if L 2 ≥ Ln := 2π √2 n [84]. By virtue of the periodic boundary conditions (3.306), the topological charge of the sphaleron φn = β + i γ 2k b(k) sn[b(k)γτ; k], (3.309) √ with (β, γ) = (0, 1), ( 1 , 3 ) or (1, √3) is zero. In fact, the configuration (3.309) − 2 2 can be interpreted as a chain of n kinks and n antikinks, alternating and equally spaced around the circle [84]. The energy of the sphaleron is L Z n o E = dτ φ˙ 2 + V(φ, φ¯ ) (3.310) | | 0 and, e.g., for the case a) in (3.305) we obtain

2n h 2 2 2 i E(φn) = 8(1 + k ) E(k) (1 k )(5 + 3k ) K(k) , (3.311) 3 √2 − − where K(k) and E(k) are the complete elliptic integrals of the first and second kind, respectively [84]. Substituting the non-BPS solutions (3.309) into (3.88), (3.156)-(3.158) or (3.182)- (3.184), we obtain a finite-action Yang-Mills configuration which is interpreted as a chain of n instanton-anti-instanton pairs sitting on S1 G/H, where G/H is × a six-dimensional nearly Kahler¨ coset space. We note that the last preceding sections deal with the special sorts of solutions of the Yang-Mills equations on the manifolds over the nearly Kahler¨ manifolds. The general case with different φ1, φ2 and φ3 can be considered by constructing a numerical solution. Finding of an analytical solution to the e.o.m. for the general case is rather problematical because of the non-linearity and derivatives of the second order. 108 CHAPTER 3. YANG-MILLS-THEORY ON NEARLY KAHLER¨ MANIFOLDS

Dyonic solutions

In the ﬁnal step we consider dyonic solutions of (3.302) characterized by introducing Lorentzian metric on R G/H. That is done by changing the signature × of the τ-coordinate of R, i.e., by modiﬁcation t = iτ so that e˜0 = dt = idτ. The − − metric on R G/H is then given by × ds2 = (e˜0)2 + δ eaeb . (3.312) − ab

Since the G-invariant ansatze¨ (3.156) and (3.182) for matrices Xa do not change their form, substituting them into the Yang-Mills equations on R G/H we arrive × at the same second-order diﬀerential equations as in the Euclidean case, except for the replacement

2 d φi φ¨ i and hence, V(φ1, φ2, φ3) V(φ1, φ2, φ3) (3.313) −→ − dt2 −→ − for the kinetic terms and the potential energy. This alteration implies a sign change of the left hand side relative to the right hand side in (3.263) and (3.298), (3.299). Moreover, the Lorentzian variant of equations (3.305) can be not derived from ﬁrst-order equations. Nevertheless, they can be integrated explicitly so that we obtain as a solution

1 γt φ(t) = β + i √2 γ cosh− , (3.314) √2

√ with (β, γ) = (0, 1), ( 1 , 3 ) or (1, √3). Any such configuration is a bounce in − 2 2 an inverted double-well potential, which from the asymptotic local minimum briefly explores the unstable region. After inserting (3.314) into the gauge potential (3.88), we arrive at dyon- type configurations with smooth non-vanishing ‘electric’ and ‘magnetic’ field strength F0a and Fab, respectively. The full energy, given by

tr(2F0aF0a + F F ) Vol(G/H), (3.315) − ab ab × for these conﬁgurations is ﬁnite, while the action is not. To the Lorentzian equations for φ1, φ2, φ3 one can also seek a numerical solutions. However, we leave that task to the prospective research papers and theses. Appendix A

Appendix

A.1 Jacobi identities

As elements of a Lie algebra the generators IA satisfy the Jacobi identity

[Ib, [Ij, Ia]] + [Ij, [Ia, Ib]] + [Ia, [Ib, Ij]] = 0. (A.1)

The commutation relation between the elements

k c i c [Ii, Ij] = fijIk, [Ii, Ia] = fiaIc, [Ia, Ib] = fabIi + fabIc (A.2) allow us to rewrite it in terms of the structure constants:

c d c d c d c i k i c i ( f f + f f f f )I + ( f f + f f f f )Ii = 0. (A.3) ja bc ab jc − jb ac d ja bc ab jk − jb ac

Since the metric has the simple form gAB = δAB, we can pull down the indices. Moreover, the generators IA are linear independent so that each of both summands in (A.3) has to vanish. Thus,

fjac fdbc + fabc fjcd + fjbc fadc = 0, fjac fibc + fkb fijk + fjbc fiac = 0. (A.4)

A.2 Jacobi elliptic functions

The Jacobi elliptic functions arise from the inversion of the elliptic integral of the ﬁrst kind, Zξ dx u = F(ξ, k) = , 0 k2 < 1, (A.5) √1 k2 sin x ≤ 0 − where k = mod u is the elliptic modulus and ξ = am(u, k) = am(u) is the Jacobi amplitude, given by 1 ξ = F− (u, k) = am(u, k). (A.6)

109 110 APPENDIX A. APPENDIX

Then the three basic functions sn, cn and dn are deﬁned by

sn[u; k] = sin(am(u, k)) = sin ξ, (A.7) cn[u; k] = cos(am(u, k)) = cos ξ, (A.8) q p dn[u; k] = 1 k2 sin2(am(u, k)) = 1 k2 sin2 ξ. (A.9) − − These functions are periodic in K(k) and K˜(k),

sn[u + 2mK + 2niK˜; k] = ( 1)msn[u; k], (A.10) − cn[u + 2mK + 2niK˜; k] = ( 1)m+ncn[u; k], (A.11) − dn[u + 2mK + 2niK˜; k] = ( 1)ndn[u; k], (A.12) − where K˜(k):= K( √1 k2) and K(k) is the complete elliptic integral of the ﬁrst π − kind, K(k):= F( 2 , k). In the following we eventually omit the parameter k and write sn[u; k] = sn(u) etc. The Jacobi elliptic functions satisfy some identities, including

sn2u + cn2u = 1, (A.13) k2sn2u + dn2u = 1, (A.14) cn2u + √1 k2 sn2u = 1 (A.15) − and

sn[u; 0] = sin u, (A.16) cn[u; 0] = cos u, (A.17) dn[u; 0] = 1, (A.18) so they are a generalization of the trigonometric functions. One may also deﬁne cn, dn and sn as solutions to the diﬀerential equations

d2 y = (2 k)2 y + y3, (A.19) dx2 − d2 y = (1 2k2)y + 2k2 y3, (A.20) dx2 − − d2 y = (1 + k2)y + 2k2 y3, (A.21) dx2 − respectively. List of Tables

2.1 Multiplication table for octonions [5]...... 25 2.2 Multiplication table for octonions [5]...... 25 2.3 Cliﬀord Algebras Clp,q, p + q < 8...... 33

3.1 Riemannian cones with special holonomy...... 79

111

Bibliography

[1] M. Kawasaki and T. Yanagida. Constraint on Cosmic Density of the String Moduli Field in Gauge-Mediated Supersymmetry-Breaking Theories. arXiv:hep-ph/9701346v3, 26 Jan 1997. [cited at p. 3]

[2] D. Harland, T. A. Ivanova, O. Lechtenfeld, and A. D. Popov. Yang-Mills Flows on Nearly K¨ahlerManifolds and G2-Instantons. arXiv:0909.2730v1 [hep-th], 15 Sep 2009. [cited at p. 3, 80, 81, 84, 94, 99, 100, 101, 102, 103, 104, 105]

[3] M. A. Shifman. Instantons in Gauge Theories. World Scientiﬁc Publishing Co. Pte. Ltd, 1994. [cited at p. 3]

[4] G. Lugo. Diﬀerential Geometry in Physics. Lecture Notes. University of North Carolina at Wilmington, 1998. [cited at p. 8]

[5] wikipedia.org. [cited at p. 8, 12, 13, 14, 15, 16, 19, 20, 22, 25, 26, 29, 31, 32, 33, 34, 35, 38, 39, 40, 41, 47, 48, 49, 50, 53, 54, 56, 58, 60, 63, 97, 111]

[6] B. A. Dubrovin, A. T. Fomenko, and S. P. Novikov. Modern Geometry. Methods and Applications. Part I. The Geometry of Surfaces, Transformation Groups, and Fields. Graduate Texts in Mathematics. Springer-Verlag New York, second edition, 1992. [cited at p. 8, 12, 14, 15, 16, 17, 20, 23, 42, 52, 54]

[7] P. W. Michor. Foundations of Diﬀerential Geometry. Lecture Notes. Institut fur¨ Mathe- matik der Universitat¨ Wien, 1997. [cited at p. 12, 13]

[8] J. M. Lee. Diﬀerential and Physical Geometry. Lecture Notes. [cited at p. 12, 16, 29, 30, 31, 33, 39]

[9] I. Kola´r,ˇ P.W. Michor, and J. Slovak.´ Natural Operations in Diﬀerential Geometry. Orig- inally published by Springer-Verlag, Berlin Heidelberg, Institut fur¨ Mathematik der Universitat¨ Wien, electronic edition, 1993. [cited at p. 12, 13]

[10] mathworld.wolfram.com. [cited at p. 12]

[11] R. O. Wells. Diﬀerential Analysis on Complex Manifolds. Graduate Texts in Mathe- matics 65. Springer Science + Business Media, LLC, third edition, 2008. [cited at p. 14, 15]

113 114 BIBLIOGRAPHY

[12] S. Helgason. Diﬀerential Geometry, Lie Groups, and Symmetric Spaces, volume Pure and Applied Mathematics of A Series of Monographs and Textbooks. Academic Press, Harcourt Brace Jovanovich, Publishers, 1978. [cited at p. 14, 46, 47, 55, 56, 59, 60]

[13] S. S. Chern, H. Chen, and K. S. Lam. Lectures on Diﬀerential Geometry, volume 1 of Series on University Mathematics. World Scientiﬁc, Singapore, 2000. [cited at p. 14, 29]

[14] M. Nakahara. Geometry, Topology and Physics. Graduate Student Series In Physics. Institute of Physics Publishing, Bristol and Philadelphia, second edition, 2003. [cited at p. 18, 54, 61, 65, 97]

[15] A. I. Kostrikin and I. R. Shafarevich (Eds.). Algebra I. Basic Notions of Algebra, volume 11 of Encyclopedia of Mathematical Science. Springer-Verlag. [cited at p. 19, 29, 56]

[16] A. P. Efremov. Quaternions: Algebra, Geometry And Physical Theories. Hypercomplex numbers in geometry and physics, 1, 2004. [cited at p. 22]

[17] J. H. Conway and D. A. Smith. On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry. A. K. Peters, Natick, Massachusetts, 2003. [cited at p. 25, 26]

[18] J. C. Baez. On Quaternions and Octonions: Their Geometry, Arithmetic, and Symmetry by John H. Conway and Derek A. Smith. Department of Mathematics, University of California, August 12, 2004. [cited at p. 25]

[19] S. Okubo. Introduction to Octonions and Other Non-associative Algebras in Physics. Montroll Memorial Lecture Series in Mathematical Physics. Cambridge University Press, 1995. [cited at p. 25]

[20] A.-K. Khaled. Beyond Octonions. arXiv:math-ph/0002023v2, 10 Feb 2000. [cited at p. 26]

[21] J. C. Baez. The Octonions, volume 39, Number 2. A Bulettin of the American Mathematical Society, Department of Mathematics, University of California, 2001. [cited at p. 26]

[22] M. O. Katanev. Geometrical Methods in Mathematical Physics. Lecture Notes. 2007/2008. [cited at p. 29]

[23] J. Gallier. Cliﬀord Algebras, Cliﬀord Groups, and a Generalization of the Quaternions: The Pin and Spin Groups. Department of Computer and Information Science University of Pennsylvania, 2009. [cited at p. 30, 31, 32, 33, 34, 39]

[24] S. Lang. Algebra. Graduate Texts in Mathematics 211. Springer-Verlag New York, Inc., 2002. [cited at p. 30]

[25] J. Snigg. Cliﬀord Algebra. A Computational Tool for Physicists. Oxford University Press, 1997. [cited at p. 31, 35, 36]

[26] I. Porteous. Cliﬀord Algebras and the Classical Groups. Cambridge Studies in Ad- vanced Mathematics. University Press, Cambridge, 1995. [cited at p. 31, 39]

[27] J. Pearson. 1,2,4,8: Cliﬀord Algebras, Bott Periodicity, and the K-Theory of the Spheres. Lecture Notes. [cited at p. 32, 33, 34, 35, 36] BIBLIOGRAPHY 115

[28] A.H. Bilge, S¸. Koc¸ak, and S. Uguz.˘ Canonical Bases for real Representations of Cliﬀord Algebras. arXiv:math-ph/0401014v2, 2 Feb 2006. [cited at p. 33, 35, 36]

[29] P. Lounesto. Cliﬀord Algebras and Spinors. London Mathematical Society Lecture Note Series 286. Cambridge University Press, second edition, 2001. [cited at p. 33, 39]

[30] S. Okubo. Representations of Cliﬀord Algebras and its Applications. arXiv:hep- th/9408165 v1, 29 Aug 1994. [cited at p. 36]

[31] P. Woit. Topics in Representation Theory: The Spinor Representation. Lecture Notes. Department of Mathematics Columbia University, 2003. [cited at p. 38]

[32] M. Wolf. On Topologically Nontrivial Field Conﬁgurations in Noncommutative Field Theories. Diploma Thesis. Dresden-Hanover 2003. [cited at p. 38]

[33] C. Chevalley. The Algebraic Theory of Spinors and Cliﬀord Algebras. Collected Works, volume 2. Springer-Verlag, 1997. [cited at p. 38]

[34] F. Cooper, A. Khare, and U. Sukhatme. Supersymmetry and Quantum Mechanics. Physics Reports 251 (995) 267-385. Elsevier Science B.V., 1995. [cited at p. 39]

[35] H. P. Nilles. Supersymmetry, Supergravity and Particle Physics. Physics Re- ports,Review Section of Physics Letters) 110, No. 1, 2 (1984) 1-162. North-Holland Physics Publishing Division, 1995. [cited at p. 39, 40]

[36] M. R. Douglas. The Geometry of String Theory. Strings And Geometry. Proceedings of the Clay Mathematics Institute 2002 Summer School on Strings and Geometry. Isaac Newton Institute Cambridge, United Kingdom, March 24April 20, 2002. 1-31. [cited at p. 39]

[37] P. West. Introduction to Supersymmetry and Supergravity. Mir, Moscow, 1989. [cited at p. 39, 40]

[38] P. Fayet and S. Ferrara. Supersymmetry. Physics report(Section C of Physics Let- ters) 32, No. 5 (1977) 249-334. North-Holland Publishing Company, Amsterdam. [cited at p. 40]

[39] D. G. C. McKeon and T. N. Sherry. Aspects of the Supersymmetry Algebra in Four Dimensional Euclidean Space. arXiv:hep-th/9810118v1, 15 Oct 1998. [cited at p. 40]

[40] J. E. Humphreys. Introduction to Lie Algebras and Representation Theory. Springer- Verlag, 1972. [cited at p. 41, 48]

[41] H. Samelson. Notes on Lie Algebras. Universitext. Springer-Verlag, third corrected edition, 1989. [cited at p. 43, 44, 46, 52]

[42] A. Skorobogatov. Lie Algebras. Lecture Notes. March 20, 2007. [cited at p. 44, 45, 46, 47, 48]

[43] S. Sternberg. Lie Algebras. Lecture Notes. April 23, 2004. [cited at p. 44, 48, 58]

[44] Planetmath.org. [cited at p. 45]

[45] J.-P. Serr. Lie Algebras and Lie Groups. Mir, Moscow, 1969. [cited at p. 47, 52, 53, 55] 116 BIBLIOGRAPHY

[46] J. T. Ottensen. Inﬁnite-dimensional Groups and Algebras in Quantum Physics. Springer- Verlag, 1995. [cited at p. 49]

[47] V. G. Kac. Inﬁnite Dimensional Lie Algebras. An Introduction. Birkhauser¨ Boston, Inc., 1983. [cited at p. 50]

[48] P. Goddard and D. Olive. Kac-Moody and Virasoro Algebras. A Reprint Volume for Physicists, volume 3 of Advanced Series in Mathematical Physics. World Scientiﬁc Publishing, Singapore, 1988. [cited at p. 50]

[49] B. C. Hall. Lie Groups, Lie Algebras, and Representations. An Elementary Introduction. Springer-Verlag New York, 2004. [cited at p. 52]

[50] V.Paulsen. An Introduction to the Theory of TopologicalGroups and their Representations. Lecture Notes. [cited at p. 54]

[51] W. Y. Hsiang. Lectures on Lie groups, volume 2 of Series on University Mathematics. World Scientiﬁc Publishing Singapore, 2000. [cited at p. 54]

[52] M. F. Atiyah, R. Bott, S. Helgason, D. Kazhdan, B. Kostant, G. Lusztig, I. G. Mac- donald, G. W. Mackey, W. Schmid, and D. J. Simms. Representation Theory of Lie Groups. London Mathematical Society, Lecture Note Series. Cambridge University Press, 1979. [cited at p. 55]

[53] A. Arvanitoyeorgos. An Introduction to Lie Groups and the Geometry of Homogeneous Spaces, volume 22 of Student Mathematical Library. Oxford University Press, illus- trated edition, 2003. [cited at p. 56]

[54] M. Paredes and S. Pizon.´ Curvature on Reductive Homogeneous Spaces, volume 41 of Revista Colombiana de Matem´aticas. 2007. [cited at p. 57, 61]

[55] W. Kuhnel.¨ Lie-Gruppen. Manuskript zur Vorlesung. Institut fur¨ Geometrie und Topologie Universitat¨ Stuttgart. [cited at p. 57]

[56] P. Koerber, D. Lust,¨ and D. Tsimpis. Type IIA AdS4 Compactiﬁcations on Cosets, Interpolations and Domain Walls. hep-th, arXiv:0804.0614v2, 2 Jul 2008. [cited at p. 57]

[57] A. A. Sagle and R. E. Walde. Introduction to Lie Groups and Lie Algebras, volume 51 of Pure and Applied Mathematics. A Series of Monographs and Textbooks. Academic Press, New York and London, 1973. [cited at p. 57, 60, 61]

[58] S. Kobayashi and K. Nomizu. Foundations of Diﬀerential Geometry, volume I, II. Nauka, Moscow, 1981. [cited at p. 58, 59, 61, 65]

[59] C. J. Isham. Modern Differential Geometry for Physicists, volume 61 of World Sci- entific Lecture Notes in Physics. World Scientific Publishing, second edition, 2001. [cited at p. 58]

[60] L. Kristensen. Invariant Metrics in Coset Spaces. Math. Scand. 6, 1958. 33-36. [cited at p. 59]

[61] A. Borel. K¨ahlerianCoset Spaces of Semi-simple Lie Groups, volume 40 of Mathematics. Proc. N. A. S, 1954. [cited at p. 61, 62] BIBLIOGRAPHY 117

[62] P.-A. Nagy. Nearly K¨ahler Geometry and Riemannian Foliations. arXiv:math/0203038v1, 5 Mar 2002. [cited at p. 63]

[63] M. Verbitsky. An Intrinsic Volume Functional on Almost Complex 6-manifolds and Nearly K¨ahlerGeometry. arXiv:math/0507179v7,2 Nov 2007. [cited at p. 63, 77, 78, 79]

[64] P.-A. Nagy. On Nearly K¨ahler Geometry. arXiv:math/0110065v1, 5 Oct 2001. [cited at p. 63]

[65] J.-B. Butruille. Homogeneous Nearly K¨ahlerManifolds. arXiv:math/0612655v1, 21 Dec 2006. [cited at p. 63]

[66] A. Collinucci and A. Wijns. Topology of Fibre Bundles and Global Aspects of Gauge Theories. arXiv:hep-th/0611201v1, 17 Nov. 2006. [cited at p. 65]

[67] M. Daniel and C. M. Viallet. The Geometrical Setting of Gauge Theories of the Yang-Mills Type. Reviews of Modern Physics, Vol. 52, No. 1, January 1980. 175-197. [cited at p. 65]

[68] K. Nomizu. Lie Groups and Diﬀerential Geometry. The Mathematical Society Of Japan, 1956. [cited at p. 65]

[69] T. Eguchi, P. B. Gilkey, and A. J. Hanson. Gravitation, Gauge Theories and Diﬀerential Geometry, volume 66. No. 6 (1980) 213393. North-Holland Publishing Company, Amsterdam, Physics Reports (Review Section of Physics Letters). [cited at p. 65]

[70] T. A. Ivanova, O. Lechtenfeld, A. D. Popov, and T. Rahn. Instantons and Yang-Mills Flows on Coset Spaces. arXive:0904.0654v1 [hep-th], 3 Apr. 2009. [cited at p. 72, 73]

[71] T. Rahn. Yang-Mills Conﬁgurations on Coset Spaces. Diploma Thesis. Universitat¨ Hannover, June 12th 2009. [cited at p. 72, 73]

[72] A. D. Popov. Hermitian-Yang-Mills Equations and Pseudo-Holomorphic Bundles on Nearly K¨ahlerand Nearly Calabi-Yau Twistor 6-manifolds. arXiv:0907.0106v2 [hep-th], 19 Aug 2009. [cited at p. 73, 76]

[73] G. Tian. Gauge Theory and Calibrated Geometry, I. Annals of Mathematics, 151 (2000). 193-268. [cited at p. 73]

[74] T. A. Ivanova and O. Lechtenfeld. Noncommutative Multi-Instantons on R2n S2. × arXiv:hep-th/0305195v2, 16 Jun 2003. [cited at p. 73]

[75] T. A. Ivanova and O. Lechtenfeld. Noncommutative Instantons on CPn. arXiv:hep- th/0603125v2, 1 Dec 2006. [cited at p. 73, 74]

[76] F. P. Correia. Hermitian Yang-Mills Instantons on Calabi-Yau Cones. JHEP12(2009)004. [cited at p. 77]

[77] D. D. Joyce. Compact Manifolds with Special Holonomy. Oxford Mathematical Mono- graphs. Oxford University Press, 2000. [cited at p. 77]

[78] Sp. Karigiannis. Deformations of G2 and Spin(7) Structures on Manifolds. April 2003. [cited at p. 78]

[79] Sp. Karigiannis. Desingularization of G2 Manifolds with Isolated Conical Singularities. arXiv:0807.3346v2 [math.AP], 23 Oct 2008. [cited at p. 78] 118 BIBLIOGRAPHY

[80] A. Moroianu and L. Ornea. Conformally Einstein Products and Nearly K¨ahlerManifolds. hal-00108146, version 4 - 5 Dec 2008. [cited at p. 78, 79]

[81] V. Cortes,´ T. Leistner, L. Schafer,¨ and F. Schulte-Hengesbach. Half-ﬂat Structures and Special Holonomy. arXiv:0907.1222v1 [math.DG], 7 Jul 2009. [cited at p. 79]

[82] C. Caviezel, P. Koerber, S. Kors,¨ D. Lust,¨ D. Tsimpis, and M. Zagermann. The Eﬀective Theory of Type IIA AdS4 Compactiﬁcations on Nilmanifolds and Cosets. arXiv:0806.3458v2 [hep-th], 11 Jul 2008. [cited at p. 85, 86, 88, 89, 92]

[83] T. A. Ivanova and O. Lechtenfeld. Yang-Mills Instantons and Dyons on Group Mani- folds. Phys. Lett. B 670 (2008) 91 (arXiv:0806.0394 [hep-th]). [cited at p. 100]

[84] N. S. Manton and T. M. Samols. Sphalerons on a Circle. Phys. Lett. B 207 (1988) 179. [cited at p. 107]